# Archive of seminars

Information on all Analysis and PDE seminars

## Main content

## 2024 (Spring)

**Date and time**: Thursday, January 25, at 14.15

**Place**: Aud. Sigma

**Speaker**: Irina Markina, Professor, Department of Mathematics, UiB

**Title**: KdV equation as a minimising L^2-energy equation.

**Abstract**: This talk is oriented to two parts of the Analysis and PDE group: the group with an interest in differential geometry and the group that is interested in non-linear partial differential equations. The motion of a rigid body in 3-D space is successfully described as a motion in the group of Euclidean transformations (rotations and translations) by making use of the Euler angles. V. Arnold proposed to describe fluid motion by replacing the finite-dimensional group of Euclidean transformations with the infinite-dimensional group of diffeomorphic transformations of a suitable space. We will consider the simplest infinite dimensional group, which is the group of diffeomorphism Diff(S) of the unit circle S, which corresponds to the description of periodic solutions of one variable. I will define the group (slightly different from Diff(S)), its Lie algebra, the metric on it (or the energy) and finally show that the equation describing the geodesics (the curves minimizing the energy) is the famous KdV equation in the fluid mechanics.

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.

**Date and time**: Thursday, February 1, at 14.15

**Place**: Aud. Sigma

**Speaker**: René Langøen, PhD. student, Department of Mathematics, UiB

**Title**: Stokes graphs of a quadratic differential related to a Rabi model

**Abstract**: To study the behaviour of solutions to a second-order linear differential equation 𝑦″+𝑄(𝑧,𝑡)𝑦=0 one can associate the quadratic differential 𝑄(𝑧)𝑑𝑧^{2} on the punctured Riemann sphere and consider its Stokes graph. We consider an ODE related to a Rabi problem describing a light-atom interaction in physics. The associated quadratic differential is meromorphic with two finite poles. The integrability condition for this type of ODE under isomonodromic deformations is related to a non-linear second-order differential equation, known as Painlevé V. In my talk, I will explain a classification of the Stokes graphs according to the nature of the zeros of the meromorphic quadratic differential originated in the Rabi model. This is a joint work with I. Markina (University of Bergen) and A. Solynin (Texas Tech, USA).

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.

**Date and time**: Thursday, February 8, at 14.15

**Place**: Aud. Sigma

**Speaker**: Martin Oen Paulsen, PhD. student, Department of Mathematics, UiB

**Title**: On the square root of a Laplace-Beltrami operator

**Abstract**: A central part of the study of free boundary problems is related to the properties of the Dirichlet to Neumann operator (DNO). In particular, it is known that DNO is related to the Beltrami-Laplace operator associated with the surface. An important application of this observation was given in the celebrated paper by Lannes published in the Journal of the AMS in 2005, where he proved the well-posedness of the water waves equations.

In this talk, I will present this key observation that relates the DNO with the square root of the Laplace-Beltrami operator.

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.

**Date and time**: Thursday, February 15, at 14.15

**Place**: Aud. Sigma

**Speaker**: Erlend Grong, Researcher, Department of Mathematics, UiB

**Title**: Infinite dimensional control on manifolds with boundary

**Abstract**: The talk will relate to controllability; the ability to reach a certain point in a space given a permissible set of movements.Though there are several standard results for finite-dimensional manifolds, controllability in infinite dimensional spaces are less understood. We are going to focus on the special case of diffeomorphism groups. In 2009, Agrachev and Caponigro described how we could look at how flows of vector fields move points around on a manifold and determine from such observations which diffeomorphisms can be generated by similar flows. We want to consider the same problem, but now allow the manifold to have boundary, and even corners in some special cases. We will both describe the Lie group structure for such groups as well as controllability result. The results presented are based on joint work with Alexander Schmeding (NTNU).

**Date and time**: Thursday, March 7, at 14.15

**Place**: Aud. Sigma

**Speaker**: Jonatan Stava, PhD.-student, Department of Mathematics, UiB

**Title**: What is the best way to understand the Bianchi Identities?

**Abstract**: The Bianchi identities are relations involving the curvature and torsion that will always be satisfied for a smooth manifold with an affine connection. These identities are equivalent to the Jacobi identity with respect to the skew-symmetrization of the tensor product in the free tensor algebra over the smooth vector fields. To get this equivalence we must include the connection to make the tensor algebra into what is called a D-algebra. It is also known that the Bianchi identities can be derived from the canonical principal connection and the canonical solder form of the frame bundle.

In this presentation we will be in two parts:

- First we look at the frame bundle. What structure is induced on the frame bundle from the connection on the underlying manifold? The goal is to get a good understanding of the curvature, torsion and the Bianchi identities on the frame bundle.
- Secondly we will see what the D-algebra is and how the Bianchi identities can be represented here. We might speculate on the relation between the frame bundle and the D-algebra.

This talk is based on early stage of a potential research project.

**Keywords:** Bianchi identities, connection, frame bundle, D-algebra

**Date and time**: Thursday, March 21, at 14.15

**Place**: Aud. 4

**Speaker**: Martin Oen Paulsen, PhD.-student, Department of Mathematics, UiB

**Title**: The Nash-Moser Theorem and Applications

**Date and time**: Thursday, April 4, at 14.15

**Place**: Aud. Sigma

**Speaker**: Francesco Ballerin, PhD Candidate, Department of Mathematics, UiB

**Title: **SO(3)-equivariant neural networks for meteorological predictions

**Abstract**: Geometric deep learning is a field that extends traditional deep learning methods to data with an underlying geometric structure, such as graphs and manifolds. It equips neural networks with mechanisms to handle non-Euclidean data and force either invariance or equivariance depending on the type of problem. This means that the result of a prediction does not depend on the symmetries of the underlining geometrical space: rotating a picture of a cat does not produce a picture of a dog and changing atlas for a manifold does not change the manifold. This tackles some of the problems of the black-box nature of neural networks forcing some useful mathematical constraints. We explore the ideas behind geometric deep learning, and we see an application to meteorological data prediction.

**Date and time**: Thursday, April 11, at 14.15

**Place**: Aud. Sigma

**Speaker**: Torunn Stavland Jensen, Master student, Department of Mathematics, UiB

**Title: **Unique Continuation for the Schrödinger Equation

**Abstract**: The first part of the talk will be a short history on unique continuation, Carleman estimates and how Hardy’s Uncertainty Principle is related to a unique continuation result for the free Schrödinger equation. In a series of work by Escauriaza, Kenig, Ponce and Vega this result was extended to the Schrödinger equation with potential and to the NLS. I will present the main steps of the proof for this result for the Schrödinger equation with potential. The proof is based on Carleman estimates, which formally is based on calculus and convexity arguments. However, going from a formal level to a rigorous one is not straight forward, and if we do not justify the computations rigorously, we can prove wrong results. In particular I will present an example of a formal Carleman argument for which the corresponding inequalities leads to a false statement.

**Date and time**: Thursday, April 18, at 14.15

**Place**: Aud. Sigma

**Speaker**: Sylvie Vega-Molino, Postdoc, Department of Mathematics, UiB

**Title: **Controllability of Shapes through Landmark Manifolds

**Abstract**: Landmark manifolds consist of distinct points that are often used to describe shapes. We show that in the Euclidean space, we can preselect two vector fields such that their flows will be able to take any n-landmark to another, regardless of the number of points n. This is a joint work with Erlend Grong.

**Date and time**: Thursday, April 25, at 14.15

**Place**: Aud. Sigma

**Speaker**: Mihaela Ifrim, Associate Professor, from University of Wisconsin Madison.

**Title: **The small data global well-posedness conjecture for 1D defocusing dispersive flows

**Abstract**: I will present a very recent conjecture which broadly asserts that small data should yield global solutions for 1D defocusing dispersive flows with cubic nonlinearities, in both semilinear and quasilinear settings. This conjecture was recently proved in several settings in joint work with Daniel Tataru.

**Date and time**: Friday, May 10, at 10.15

**Place**: Aud. Sigma

**Speaker**: Marcos Salvai, Professor from National University of Cordoba, Argentina.

**Title: **The sub-Riemannian geometry of screw motions with constant pitch

**Abstract**: Let M be an oriented three-dimensional Riemannian manifold of constant sectional curvature k = 0 , 1, -1 and let SO(M) be its direct orthonormal frame bundle (direct refers to positive orientation), which has dimension six and may be thought of as the set of all positions of a small body in M. Given λ ∈ R, there is a three-dimensional distribution D^λ on SO(M) accounting for infinitesimal rototranslations of constant pitch λ. When λ ≠ k^2, there is a canonical sub-Riemannian structure on D^λ. We describe its geodesics. For k = 0, -1, we compute the lengths of all periodic geodesics of ( SO(M) , D^λ ) in terms of the lengths and the holonomies of the periodic geodesics of M, when M has positive injectivity radius.

It turns out that the notion of rototranslating with constant pitch makes sense for some higher dimensional Riemannian manifolds, for instance, for R^7 via the octonionic cross product, or for compact Lie groups. We define sub-Riemannian structures analogous to the above and find some of their geodesis.

**Date and time**: Thursday, May 16, at 14.15

**Place**: Aud. Sigma

**Speaker**: Hans Z. Munthe-Kaas, Professor, Department of Mathematics, UiB.

**Title: **Rough paths on manifolds

**Abstract**: This work studies rough differential equations (RDEs) on homogeneous spaces. We provide an explicit expansion of the solution at each point of the real line using decorated planar forests. The notion of planarly branched rough path is developed, following Gubinelli's branched rough paths. The main difference being the replacement of the Butcher–Connes–Kreimer Hopf algebra of non-planar rooted trees by the Munthe-Kaas–Wright Hopf algebra of planar rooted forests. The latter underlies the extension of Butcher's *B*-series to Lie–Butcher series known in Lie group integration theory. Planarly branched rough paths admit the study of RDEs on homogeneous spaces, the same way Gubinelli's branched rough paths are used for RDEs on finite-dimensional vector spaces. An analogue of Lyons' extension theorem is proven. Under analyticity assumptions on the coefficients and when the Hölder index of the driving path is one, we show convergence of the planar forest expansion in a small time interval.

**Date and time**: Monday, June 3, at 10.00

**Place**: Aud. Sigma

**Speaker**: Gustavo Ponce, Professor, Department of Mathematics, University of California, Santa Barbara, USA

**Title: **Unique Continuation Properties of Nonlocal Nonlinear Dispersive Models.

**Abstract**: We shall study unique continuation properties for a class of non-local “dispersive" models.

We shall consider two type of unique continuation problems: local and asymptotic to infinity.

We shall review several results for several equations including the Benjamin-Ono equation, the ILW equation., the Camassa-Holm equation, and the Benjamin-Bona-Mahony equation.

**Date and time**: Monday, June 3, at 11.00

**Place**: Aud. Sigma

**Speaker**: Luis Vega, Professor, BCAM, University of the Basque Country, Spain

**Title: **The binormal flow and the evolution of viscous vortex filaments.

**Abstract**: I will present the so called Localized Induction Approximation that describes the dynamics of a vortex filament according to the Binormal Curvature Flow (BF). I’ll give a result about the desingularization of the Biot-Savart integral proved with Marco A. Fontelos within the framework of Navier-Stokes equations. Some particular examples regarding BF obtained with Valeria Banica will be also considered. These examples allow to connect BF with the so-called Riemann non-differentiable function and the Frisch-Parisi approach to turbulence.

**Date and time**: Friday, June 7, at 14.15

**Place**: Delta (4A9f), Realfagbygget

**Speaker**: Jonatan Stava, PhD. candidate @ the University of Bergen

**Title: **Nomizu's canonical connection of the second kind

**Abstract**: Runge-Kutta methods is a common class of methods for solving initial value problems in R^n. These methods approximate the exact flow by combining curves with constant velocity, or just "straight lines". If we want to general these methods to manifolds other than R^n we must use the equivalent of a constant velocity curve which is called a geodesic, but these curves are dependent on the choice of connection on the manifold. A connection is a way of taking the derivative of one vector field in the direction of another. In general, there exists an infinite number of possible connections on a manifold. We say that a connection is canonical if it is the unique connection which satisfies certain conditions which are natural for the manifold in question. For instance, on a Riemannian manifold, the Levi-Civita connection is the canonical connection which is torsion free and preserves the metric.

Homogeneous spaces are an important type of manifolds that can be represented as quotients of Lie groups. Is there a canonical connection on homogeneous spaces? This was answered by Nomizu in an article from 1954. The goals of this talk:

Give a good introduction to the topic of connections.

Prove Nomizu's theorem which relates invariant connections on a reductive homogeneous space G/H with products on a vector subspace of the Lie algebra of G.

Use this theorem to express some particularly nice connections, ending with the one that Nomizu names the canonical connection of the second kind.

## 2023 (Fall)

**Date and time**: Tuesday, November 21, at 12.15

**Place**: Aud. Sigma

**Speaker**: Didier Pilod, Associate Professor, Department of Mathematics, UiB

**Title**: Global well-posedness for the Benjamin-Ono and intermediate long wave equations in L^2

Abstract: In the first part of the talk, I will revisit the well-posedness theory of the Benjamin-Ono equation which has received a lot of attention in the past 20 years. In particular, I will explain the gauge transformation introduced by Tao in 2004, which was a breakthrough in order to attain lower regularities. In the second part of the seminar, I will summarize a joint work with Luc Molinet published in 2012, where relying on the gauge transformation of Tao, we revisit the proof of the well-posedness in L^2 both on the line and on the torus (the result on R was obtained first by Ionescu and Kenig in 2007, while the one on T was proved by Molinet in 2008). Finally, I will explain how these ideas extend to perturbations of the Benjamin-Ono equation like the Intermediate Long Wave equation. This last result was obtained very recently in collaboration with Andreia Chapouto, Guopeng Li and Tadahiro Oh (University of Edinburgh).

**Date and time**: Tuesday, November 14, at 12.15

**Place**: Aud. Sigma

**Speaker**:Jonatan Stava, Ph.D. student of the Department of Mathematics, UiB

**Title**: One (final) talk on Lie admissible triple algebras and other connection algebras

**Abstract**: Lie admissible triple algebras (LAT) appear as the connection algebras of symmetric spaces. An operator we call the triple-bracket is fundamental to describe and understand LAT. An algebra where the triple-bracket is zero is called pre-Lie, and the free pre-Lie algebra can be described by non-planar trees. LAT is a generalization of pre-Lie, hence the basis of the free LAT contains the non-planar trees, but in addition we need to account for the triple-bracket.

In this talk we present this basis and give an outline of the proof that this is indeed a basis for the free LAT, using OSBB-words which is an alternative basis for the free tensor algebra. In addition, we will give an (in some sense complete) overview of algebraic structures associated with different types of invariant manifolds.

**Date and time**: Tuesday, November 07, at 12.15

**Place**: Aud. Sigma

**Speaker**: Martin Oen Paulsen, Ph.D. student of the Department of Mathematics, UiB

**Title**: Justification of the Benjamin-Ono equation as an internal water wave model

**Abstract**: The Benjamin-Ono (BO) equation is a nonlocal asymptotic model for the unidirectional propagation of weakly nonlinear, long internal waves in a two-layer fluid. The equation was introduced formally by Benjamin in the '60s and has been a source of active research since. For instance, the study of the long-time behavior of solutions, stability of traveling waves, and the low regularity well-posedness of the initial value problem. However, despite the rich theory for the BO equation, it is still an open question whether its solutions are close to the ones of the original physical system.

In this talk, I will explain the main steps involved in the rigorous derivation of the BO equation.

**Date and time**: Tuesday, October 31 at 12.15

**Place**: Aud. Sigma

**Speaker**: Sigmund Selberg, Professor of the Department of Mathematics, UiB

**Title**: Fourier restriction theorems and space-time estimates for the wave equation

**Abstract**: I will discuss three themes which can all be related to the restriction of the Fourier transform to surfaces (spheres and cones): The Stein-Tomas theorem for the sphere, the Strichartz space-time estimate for solutions of the wave equation, and bilinear generalisations of the latter, first considered by Klainerman and Machedon.

The main focus will be on the Stein-Tomas theorem, for which I will show you my favourite (very simple) proof.

**Date and time**: Tuesday, October 24 at 12.15

**Place**: Aud. Sigma

**Speaker**: Torunn Stavland Jensen, master student, Department of Mathematics, UiB

**Title**: The Hardy Uncertainty principle for the Schrödinger equation.

**Abstract**: The Hardy Uncertainty principle states that if $f(x)=O(e^{-|x|^2/\beta^2 })$, $\hat{f}(\xi)=O(e^{-(4|\xi|^2)/\alpha^2 })$, $\alpha\beta<4$ then $f\equiv0$. This result can be rewritten in terms of the solution of the free Schrödinger equation. The original proof of the Hardy Uncertainty was based on the Phragmén-Lindelöf theorem. However, to extend the result of the Hardy Uncertainty principle to the Schrödinger equation with a potential and to the nonlinear Schrödinger equation, we need a proof that does not depend on analyticity.

I will present some of the work done by Escauriaza, Kenig, Ponce and Vega where they use the methods based on Carleman estimates to prove the Hardy uncertainty principle for the Schrödinger equation with potential. In particular, we will start with doing a formal proof for the free Schrödinger equation. Going from the formal level to a rigorous one is not straight forward. For example, if $u$ is a solution in $L^2$, it is not always true that $e^\phi u$ is in $L^2$ , for some weight $\phi$. If time allows we will also look at the main steps of the proof for the Schrödinger equation with potential.

**Date and time**: Tuesday, October 17 at 12.15

**Place**: Aud. Sigma

**Speaker**: Joao Pedro Ramos, EPFL Lausane

**Title**: Uncertainty principles and interpolation formulae in analysis and PDE.

**Abstract**: The famous Heisenberg uncertainty principle predicts that a function and its Fourier transform cannot both be too concentrated in space and frequency, unless the function vanishes identically. This principle has many classical counterparts of similar flavour in Fourier analysis, such as the Hardy uncertainty principle and the Amrein-Berthier theorem. In recent years, several breakthrough results have shed new light onto problems in the uncertainty realm. In particular, in the realm of Fourier analysis, we mention the Radchenko-Viazovska interpolation formula, which provides an explicit way to recover an even smooth function from its values - and those of its Fourier transform - at roots of integers. On the other hand, on a more PDE-based perspective, we may mention the series of papers by Escauriaza-Kenig-Ponce-Vega, which generalised the Hardy uncertainty to the Schrödinger equation setting in its sharp form. In this talk, we will go through some of these uncertainty results, from the most classical to some of the most recent. We will put particular emphasis on possible directions, open problems and conjectures in this area. (Based on joint works with M. Sousa, F. Gonçalves, M. Stoller and C. Kehle).

**Date and time**: Tuesday, October 10 at 12.15

**Place**: Aud. Sigma

**Speaker**: René Langøen, Ph.D student, Department of Mathematics, UiB

**Title**: An introduction to two-dimensional conformal field theory, the Witt algebra and the Virasoro algebra.

**Abstract**: A conformal field theory (CFT) is a field theory that is invariant under conformal transformations. Conformal field theory is a physical model for particles, where the particles are treated as an excited state of an underlying field. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations. The algebra of the vector fields, whose flow generates conformal transformations on the Riemann sphere, is the Witt algebra. When the physicists do “the quantization” they like to work with the Virasoro algebra, which is the central extension of the Witt algebra. Our goal is to derive the Witt algebra, and then explain the central extension to the Virasoro algebra.

**Date and time**: Tuesday, October 03 at 12.15

**Place**: Aud. Sigma

**Speaker**: Sylvie Vega-Molino, Postdoc, Department of Mathematics, UiB

**Title**: The Prytz Planimeter

Abstract: The Prytz planimeter is a simple mechanical device that historically was used to approximate areas of plane regions. In this talk I will present a mathematical description and analysis of the planimeter in terms of sub-Riemannian geometry, affine connections, and horizontal lifts. We will review these central concepts in differential geometry through the lens of considerations of the planimeter, and also see practical extensions of these ideas to other real-world problems.

**Date and time**: Tuesday, September 26 at 12.15

**Place**: Aud. Sigma

**Speaker**: Erlend Grong, Researcher of TMS, Department of Mathematics, UiB

**Title**: Neumann eigenvalues gap of the sub-Laplacian on Sasakian manifolds

**Abstract**: For the Laplacian on a Riemannian manifold, there is a spectral gap for positive manifolds that follows from looking at the Laplacian on forms, and using a an estimate in the style of Lichnerowicz to get a gap that depends on the manifold curvature and dimension.

However, there are also methods, started by Zhong and Yang, which take the diameter into consideration and that can work for arbitrary compact Riemannian manifolds.

The first method of spectral gap has been generalized to sub-Laplacians of sub-Riemannian manifolds, but not the latter. In the talk, I present some very early results for discussion.

**Date and time**: Tuesday, September 12 at 12.15

**Place**: Aud. Sigma

**Speaker**: Adrien Ange Andre Laurent, postdoc, Department of Mathematics, UiB

**Title**: Exotic aromatic B-series - A geometric view

Abstract: Before coming to Bergen, I introduced a new formalism of trees called exotic aromatic trees, for the computation of order conditions for the numerical approximation of stochastic differential equations. As this formalism is mainly used to deal with terrible calculations of order conditions and as I was discussing with the friendly people in pure math in Bergen, I was wondering if the exotic formalism is just a tool or if it is interesting beyond its numerical applications.In this talk, we will review the numerical applications of the exotic formalism, and we will show that exotic aromatic B-series satisfy a universal geometric property of equivariance. This universal property confirms that the exotic aromatic formalism is a natural generalization of aromatic B-series that is interesting not only from the numerical point of view. We will also give a complete classification of the subsets of exotic aromatic B-series with strong equivariance properties and shall present ongoing works involving the exotic formalism.

**Date and time**: Tuesday, September 05 at 12.15

**Place**: Aud. Sigma

**Speaker**: Francesco Ballerin, PhD student, Department of Mathematics, UiB

**Title**: Geometry of the Visual Cortex with Applications to Image Inpainting and Enhancement

Abstract Equipping the rototranslation group SE(2) with a sub-Riemannian structure inspired by the visual cortex V1, we propose algorithms for image inpainting and enhancement based on hypoelliptic diffusion. We innovate on previous implementations of the methods by Citti, Sarti and Boscain et al., by proposing an alternative that prevents fading and capable of producing sharper results in a procedure that we call WaxOn-WaxOff. We also exploit the sub-Riemannian structure to define a completely new unsharp filter using SE(2), analogous of the classical unsharp filter for 2D image processing, with applications to image enhancement. We demonstrate our method on blood vessels enhancement in retinal scans.

**Date and time**: Tuesday, August 22 at 12.15

**Place**: Aud. Sigma

**Speaker**: Bruno Carneiro da Cunha, Professor, Departamento de Física, Universidade Federal de Pernambuco, Brasil

**Title**: Liouville field theory for interface dynamics

Abstract: In physics, the boundary between two regions corresponding to different phases or constituents have interesting intrinsic dynamics. In two dimensions, this interface dynamics can be modeled using conformal maps, in an interesting application of the uniformization theorem, and of the Schwarz function associated to the boundary itself. In this seminar I will outline its construction, showing how it leads naturally to a particular case of the Liouville differential equation with a suitable boundary term, and how the generic (classical) boundary dynamics can be implemented from the action of the Virasoro algebra.

## 2023 (Spring)

**Time and place:** Thursday June 8 at 12:15 in Auditorium 4.

**Speaker:** Marco Cacace, exchange student

**Title**: Weak approximation of stochastic dynamics on Lie groups

**Abstract**: We propose a new methodology for the creation of high weak order stochastic Lie-Runge-Kutta methods for time homogenous SDE. More precisely, we establish sufficient conditions for a class of second order elliptic operators to generate a Feller semigroup on a general manifold of bounded geometry. We give an estimate of a Feller semi-group in terms of its infinitesimal generator, and we use that estimate to derive the order conditions for a weak second order Stochastic Lie Runge-Kutta method on a matrix Lie group.

**Time**: May 30th

**Speaker**:** Alexander Solynin, Texas Tech**

**Title**: Problems on the loss of heat: Herd instinct versus individual feeling

**Abstract**: /

**Time**: May 11th

**Speaker**: Erlend Grong, UiB

**Title**:Sub-Riemannian geometry, most probable paths and transformations.

**Abstract**: Doing statistics on a Riemannian manifold becomes very complicated for the reason that we lack pluss to define such things as mean and variance. Using the Riemannian distance, we can define a mean know as the Fréchet mean, but this gives no concept of asymmetry, also known as anisotropy. We introduce an alternative definition of mean called the diffusion mean, which is able to both give a mean and the analogue of a covariance matrix for a dataset on a Riemannian manifolds.Surprisingly, computing this mean and covariance is related to sub-Riemannian geometry. We describe how sub-Riemannian geometry can be applied in this setting, and mention some finite dimensional and infinite-dimensional applications.

**Time**: May 4th

**Speaker**: Morten Pederson, INRIA Sophia Antipolis/Copenhagen

**Title**: Principal subbundles for dimension reduction

**Abstract**: In this talk, we demonstrate how sub-Riemannian geometry can be used for manifold learning and surface reconstruction by combining local linear approximations of a point cloud into a lower rank tangent bundle. In particular, local approximations obtained by local PCAs are collected into a rank \textit{k} tangent subbundle on $\mathbb{R}^d$, $k<d$, denoted a principal subbundle. This determines a sub-Riemannian metric on $\R^d$. We show that sub-Riemannian geodesics w.r.t. this metric can successfully be applied to a number of important problems, such as: explicit construction of an approximating submanifold $M$, construction of a representation of the point-cloud in $\R^k$, and computation of distances between observations, taking the learned geometry into account. The reconstruction is guaranteed to equal the true submanifold in the limit case where tangent spaces are estimated exactly. Via simulations, we show that the framework is robust when applied to noisy data. Furthermore, we show that the framework generalizes to observations residing on an a priori known Riemannian manifold.

**Time**: April 27th

**Speaker**: Didier Pilod and Kundan Kumar

**Title**: On some of Luis Caffarelli's comtributions

**Abstract**: In this seminar, we will discuss two of the many contributions of Caffarelli who was recently awarded the Abel prize.In the first part of talk, Didier will present the Caffarelli-Silvestre extension formula introduced in a seminal paper published in Comm. PDE in 2007. This formula characterizes the fractional Laplacian through an extension problem to the upper-half plane by solving a (degenerate) elliptic PDE. The interest of this formula lies in the fact that one can study properties of the fractional Laplacian, which is a non-local operator, by using local techniques on the extension problem.In the second part pf the talk, Kundan will discuss the contribution(s) of Caffarelli on free boundary and/or fully nonlinear equations.

**Time**: March 23rd

**Speaker**: Gianmarco Vega-Molino, UiB

**Title**: Horizontal Holonomy of H-type Foliations

**Abstract**: In this talk I will present the notion of sub-Riemannian holonomy for H-type foliations, a class of sub-Riemannian manifolds jointly introduced with Fabrice Baudoin, Erlend Grong, and Luca Rizzi. Arising as certain sub-Riemannian geometries transverse to Riemannian foliations, they are ideally suited to study the intersection of "extrinsic" and "intrinsic" approaches to sub-Riemannian geometry. Riemannian holonomy is the study of endomorphisms of the tangent spaces induced by the parallel transport of affine connections; in a sub-Riemannian setting one can study an analogue via adapted connections.

The talk will include an introduction to these structures suitable for persons not familiar with sub-Riemannian geometry, and we will come to a theorem describing the horizontal holonomy of these spaces.

**Time**: March 16th

**Speaker**: Abdon Moutinho, Paris Nord

**Title**: On the collision problem of two kinks with low speed

Abstract:We will talk about our results on the elasticity of the collision of two kinks with low speed v>0 for the nonlinear wave equation of dimension 1+1 known as the phi^6 model. We will show that the collision of the two solitons is "almost" elastic and that, after the collision, the size of the energy norm of the remainder and the size of the defect of the speed of each soliton can be, for any k>0, of the order of any monomial v^{k} if v is small enough

**Time**: January 26th

**Speaker**: Frédéric Valet, UiB

**Title**: Collision of two solitary waves for the Zakharov-Kuznetsov equation

**Abstract**: The Zakharov-Kuznetsov (ZK) equation in dimension 2 is a generalization in plasma physics of the one- dimensional Korteweg de Vries equation (KdV). Both equations admit solitary waves, that are solutions moving in one direction at a constant velocity, vanishing at infinity in space. When two solitary waves collide, two phe- nomena can occur: either the structure of two solitary waves is conserved without any loss of energy and change of sizes (elastic collision), or the structure is lost or modified (inelastic collision). As a completely integrable equation, KdV only admits elastic collisions. The goal of this talk is to explain the collision phenomenon for two solitary waves having almost the same size for ZK, and to describe the inelasticity of the collision. The talk is based on current works with Didier Pilod.

**Time**: January 12th

**Speaker**: Stefan Le Coz, Toulouse

**Title**: Ground states on nonlinear quantum graphs

**Abstract**: The nonlinear Schrödinger equation is an ubiquitous model in physics, with numerous applications in areas as divers as Bose-Einstein condensation or nonlinear optics. In many physical situations, the underlying space is essentially one dimensional and can be modeled as a metric graph, i.e. a collection of vertices and edges with finiter or infinite lengths. The mathematical study of this type of model is very recent and is gaining an incredible momentum. In this talk, we will review some of the results concerning the existence of nonlinear Schrödinger ground states on graphs and present a numerical approach for their computation. This is a joint work with Christophe Besse and Romain Duboscq.

## 2022 (Fall)

### August

**August 25, 2022: Louis Emerald, Rennes***Title:*On the rigorous justification of full dispersion models in coastal oceanography*Abstract:*The first full dispersion model was introduced by G. Whitham in 1967 to study the Stokes waves of maximal amplitude and the wavebreaking phenomenon. It is a modification of the Korteweg-de Vries equations which have the same dispersion relation as the general water-waves model. Afterward, numerous unidirectional and bidirectional full dispersion models were introduced in the literature. In the first part of the talk, we use classical techniques on free surface elliptic equations to derive rigorously the Whitham-Boussinesq systems. In the second part, we justify rigorously Whitham's model using two different methods. One is adapted to the propagation of unidirectional waves and uses pseudo-differential calculus. The other one is adapted to the propagation of bidirectional waves. It is based on a generalisation of Birkhoff's normal form algorithm.

### October

**October 6, 2022: Gianmarco Vega-Molino, Bergen***Title:*Local Invariants and Geometry of the sub-Laplacian on H-type Foliations*Abstract:*H-type foliations are studied in the framework of sub-Riemannian geometry with bracket generating distribution defined as the bundle transversal to the fibers. Equipping the manifold with the Bott connection we consider the scalar horizontal curvature as well as a new local invariant induced from the vertical distribution. We extend recent results on the small-time asymptotics of the sub-Riemannanian heat kernel on quaternion-contact (qc-)manifolds due to A. Laaroussi and we express the second heat invariant in sub-Riemannian geometry as a linear combination of the mentioned invariant. The use of an analog to normal coordinates in Riemannian geometry that are well-adapted to the geometric structure of H-type foliations allows us to consider the pull-back of Korányi balls. We explicitly obtain the first three terms in the asymptotic expansion of their Popp volume for small radii. Finally, we address the question of when our manifold is locally isometric as a sub-Riemannian manifold to its H-type tangent group.This talk is based on a joint work with Irina Markina (Bergen), Wolfram Bauer (Hannover) and Abdellah Laaroussi (Hannover)

**October 13, 2022: Didier Pilod, Bergen**- Title: Unique continuation properties for the fractional Laplacian
- Abstract: In the first part of the talk, I will introduce the unique continuation property, and describe a classical method of proof based on Carleman estimates.
In the second part of the talk, I will review some unique continuation properties for the fractional Laplacian obtained by Rüland (CPDE 2015) and Ghosh, Salo, Uhlmann (Analysis&PDE 2020). These properties rely on the Cafarelli-Silvestre extension, allowing to describe the fractional Laplacian through a singular elliptic problem in the upper-half space, and a Carleman estimate for the extension problem obtained by Rüland.

**November**

**November 10, 2022: Gunhee Cho, UC Santa Barbra***Title:*Stochastic differential geometry and applications of probabilistic coupling methods*Abstract:*The first part discusses how stochastic differential geometry, especially coupling methods, are used in relation to the long-standing open problems of hyperbolic complex geometry, and the recent results including the stochastic Schwarz Lemma and applications.The second part examines further why the Kendall-Cranston probabilistic coupling method is potentially important and useful in studying differential geometry, and discusses the first Dirichlet eigenvalue estimate on K\"ahler manifolds, which is also a very recent result. The first part is relevant to the joint work with M. Gordina, M. Chae, and G. Yang. Also, the second part is related to the joint work with F. Baudoin, and G. Yang.

**November 10, 2022: Kenro Furutani, Osaka Metropolitan University, Osaka Central Advanced Mathematical Institute***Title:*Calabi-Yau structure and Bargmann type transformation on the Cayley projective plane

**November 17, 2022: Adrien Laurent, Bergen***Title:*The aromatic bicomplex for the study of volume-preserving integrators*Abstract:*Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that no B-series method can preserve volume in general, except for the exact solution of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this talk, we provide a complete characterization of the divergence-free aromatic forests by introducing a similar object to the variational bicomplex from differential geometry in the context of aromatic forests. The analysis provides a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator.

**December**

**December 8, 2022: Giacomo Gavelli, Bologna***Title:*Various notions of hyperbolicity and their relationship*Abstract:*The aim of the talk is to show a comparison between three different notions of hyperbolic spaces: Riemannian manifolds of constant negative curvature, Gromov-hyperbolic metric spaces and Kobayashi-hyperbolic complex spaces.In the first part of the talk I will introduce three classical hyperbolic spaces as Riemannian manifolds, which result to be isometric: the hyperboloid, the Poincaré ball and the Poincaré half-space. On these I will introduce the notions of hyperbolic segments, angles, triangles, geodesics, circles and isometries.

In the second part I will introduce Gromov and Kobayashi-hyperbolicity. According to Gromov, hyperbolic spaces are characterized by having, in a sense, slim-triangles. The classical hyperbolic spaces have this property. Kobayashi, in 1967, defined a natural semi-distance on any complex space, which is said to be Kobayashi-hyperbolic when such semi-distance is a distance. I will state a result, proven by Bonk and Balogh in 2000, which grants that bounded strictly pseudoconvex domains with C2-smooth boundary, when equipped with the Kobayashi distance, are Gromov hyperbolic.

## 2022 (Spring)

### February

**February 1, 2022: Erlend Grong, University of Bergen,***Title:*The Gauss-Bonnet theorem in sub-Riemannian geometry*Abstract:*The Gauss-Bonnet theorem connects a global invariant, the Euler characteristic, with a local invariants, the Gaussian curvature of the surface and the geodeisc curvature of the boundary. It also explains why we can find the curvature of a space by checking if the angles of a triangle add up to more or less than 180 degrees. We will give a general, low-level introduction to this result.Next, we are going to look at what happens when the inner product on the ambient space goes to infinity outside a given subbundle. Surprisingly, this shows that the Euler characteristic is only dependent on local invariants near to what is known as the characteristic set.

**February 8, 2022: Ilia Zlotnikov, University of Stavanger (Zoom)***Title:*On extreme points of the unit balls of the spaces of analytic function.

**February 15, 2022: Irina Markina, University of Bergen***Title:*On the module of measure.*Abstract:*In the talk, I want to explain one tool used in the geometric measure theory. It originated in the theory of functions of complex variables to show completeness of some functional classes and it got the name extremal lengths or module of a family of curves. It was extended to a notion in n-dimensional Euclidean spaces with generalization to the module of measure supported on k-dimensional Lipschitz manifolds.For some specific situations, the module of curves coincides with the capacity widely used to describe the exceptional sets in the Sobolev spaces. If time allows, I will touch on the recent results related to the study of modulus in some non-Euclidean geometric measure theory.

**February 22, 2022: Arnaud Eychenne, University of Bergen***Title:*Asymptotic N-soliton-like solutions of the fractional Korteweg-de Vries equation*Abstract:*In 1834, John Scott Russell discovered a soliton, which is a wave moving without deformation. Since this discovery, solitons have been studied extensively. Nowadays, solitons are natural objects in physics or biology. We will talk about the existence of N-soliton-like solutions, which are solutions that behave at infinity like a sum of N decoupled solitons, for the fractional Korteweg-de Vries equation (fKdV) $\partial_t u +\partial_x (|D|^{\alpha}u - u^2)$. The first proof of the existence of N-soliton-like solutions that is not based on the complete integrability was done around 20 years ago for local generalizations of the KdV equation. In this talk, we will explain briefly the construction of the N-soliton-like solutions and the difficulties caused by the non-locality of the operator $|D|^{\alpha}$ that we had to overcome in the case of fKdV.

### March

**March 1, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)***Title:*Geometry of Riemannian Homogeneous spaces, Lecture 1: Homogeneous Riemannian manifolds and their description

**March 8, 2022: Didier Pilod, University of Bergen***Title:*Finite point blowup for the critical generalized Korteweg-de Vries equation.*Abstract:*In the last twenty years, there have been significant advances in the study of the blow-up phenomenon for the critical generalized Korteweg-de Vries (gKdV) equation, including the determination of sufficient conditions for blowup, the stability of blowup in a refined topology and the classification of minimal mass blowup. Exotic blow-up solutions with a continuum of blow-up rates and multi-point blow-up solutions were also constructed. However, all these results, as well as numerical simulations, involve the bubbling of a solitary wave going at infinity at the blow-up time, which means that the blow-up dynamics and the residue are eventually uncoupled. Even at the formal level, there was no indication whether blowup at a finite point could occur for this equation.After reviewing the theory of blow-up for the critical gKdV equation in the first part of the talk, we will answer this question by constructing solutions that blow up in finite time under the form of a single-bubble concentrating the ground state at a finite point with an unforeseen blow-up rate.Finding a blow-up rate intermediate between the self-similar rate and other rates previously known also reopens the question of which blow-up rates are actually possible for this equation.This talk is based on a joint work with Yvan Martel (École Polytechnique/France).

**March 15, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)***Title:*Geometry of Riemannian Homogeneous spaces, Lecture 2: Some special classes of Riemannian homogeneous spaces

**March 22, 2022: Douglas Svensson Seth, NTNU***Title:*The three-dimensional steady water wave problem with vorticity*Abstract:*The water wave problem concerns solving a free boundary problem. Specifically, the equations of motion for a fluid in a two- or three-dimensional domain where the shape of the upper boundary is unknown. The problem becomes steady through the assumption that the waves travel with uniform and constant speed. The two-dimensional theory is generally more developed than the three-dimensional due to being the older of the two. Already in the mid 1800s Stokes had begun investigating the two-dimensional problem and the first rigorous existence results are due to Nekrasov and Levi-Cività (independently) in the 1920s. On the other hand, the first corresponding rigorous existence result in three dimensions took until 1981 and is due to Reeder and Shinbrot.The first part of the talk will be dedicated to a brief overview of the water wave problem in both two and three dimensions. Here we also highlight some of the differences between the two- and three-dimensional problems. The second part of the talk will be dedicated to two more recent existence results for the three-dimensional problem where the vorticity (curl of the velocity) is nonzero. In the first we assume that the velocity field of the water is a Beltrami field. In the other, the vorticity is instead given by an assumption that stems from magnetohydrodynamics. This talk is based on joint work with Erik Wahlén (Lund University), Evgeniy Lokharu (Lund University) and Kristoffer Varholm (NTNU).

**March 29, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)***Title:*Geometry of Riemannian Homogeneous spaces, Lecture 3: Geodesic orbit Riemannian spaces and some their subclasses

**April**

**April 4, 2022: Yuri Nikonorov, Russian Academy of Sciences (Zoom)***Title:*Geometry of Riemannian Homogeneous spaces, Lecture 4: The Ricci curvature and related problems

*Date and place: Tuesday April 12: Easter break***April 19, 2022: Gianmarco Vega-Molino, University of Bergen***Title:*Morse Theory, Studying Geometry and Topology through Critical Points*Abstract:*Elementary results in the calculus of functions of the real line show that we can understand the geometry of curves through the study of zeros of derivatives. Extrema of differentiable functions always occur at these critical points, and the nature of the critical points can be understood by considering the behavior of the second derivative. These notions are extended to multivariate functions by analogously considering the gradient and Hessian. In the 1920s Marston Morse initiated the study of surfaces (and more generally manifolds) by considering the critical points of functions on them, from which it is possible to determine both local and global properties; in particular, one can recover topological information such as the Euler characteristic. In this talk, I will present an introduction to Morse theory. Familiarity with differential geometry (in particular, smooth manifolds) is not a prerequisite.

**April 26, 2022: Fátima Silva Leite, Coimbra - Portugal***Title:*Interpolation on Riemannian Manifolds*Abstract:*The problem of finding a smooth curve that interpolates a set of points on a Riemannian manifold, and satisfies some boundary conditions, is particularly important in applied areas such as robotics, computer vision and medical imaging.In this seminar we start with some motivating examples and then will revisit several methods to generate interpolating Riemannian splines. Our main focus will be on a variational approach to generate splines that minimize the intrinsic acceleration, and on a method based on rolling motions that emerged to overcome difficulties in finding explicit solutions for the Euler-Lagrange equation of the previous optimization problem.

**May**

**May 3, 2022: Felipe Linares, IMPA Brazil***Title:*

**May 10, 2022: Martin Oen Paulsen, University of Bergen***Title:*Justification of a full dispersion model from the water wave system*Abstract:*A fundamental question in the derivation of an asymptotic model is whether its solution converges to the solution of the original physical system. In particular, we say that an asymptotic model is a valid approximation of the Euler equations with a free surface if we can answer the following points in the affirmative:rThe solutions of the water wave equations exist on the relevant scale.The solutions of the asymptotic model exist (at least) on the same time scale.Lastly, we must establish the consistency between the asymptotic model and the water wave equations and show that the error is "small" when comparing the two solutions.In the first part of the talk, the aim is to introduce the governing equations in water wave theory. Then discuss the three points above and explain the rigorous justification of some famous asymptotic models. The second part of the talk will focus on 2., explaining a new 'long time' well-posedness result for a Whitham-Boussinesq system.

**May 17, 2022: National Holiday****May 24: Francesco Ballerin, University of Bergen,***Tile:**Abstract:*A 2D image is perceived by the human brain through the visual cortex V1, a part of the occipital lobe which is sensitive to orientation. This sensitivity intrinsically fills-in gaps in the perceived image depending on the gradient of the image in a neighborhood, restoring corrupted portions of the field of view. The visual cortex V1 can be mathematically modeled as SE(2), a sub-Riemannian geometry which can be exploited to produce image restoration algorithms. In this talk the current state-of-the-art regarding image restoration models based on SE(2) is presented and a proposed new algorithm is introduced.

- May 31: No seminar. Abel prize lecture by Lázió Lovász.

**June**

- June 20: René Langøen, University of Bergen
*Title:*The direct monodromy problem and isomonodromic deformations for the Rabi model.*Abstract:*We discuss the local and global solutions of the Rabi model in Garnier form, a linear system of first order differential equations, with complex rational coefficients. The analytic continuation of the local solutions are described by a monodromy group, which gives a matrix representation of the fundamental group of the punctured Riemann sphere. A detailed geometric description of linear systems of first order differential equations is given, in terms of a local family of connection forms on a principal bundle. The geometric description reveals the Frobenius integrability conditions, which are used to obtain necessary and sufficient conditions for an isomonodromic deformation of the Rabi model.

**2021 (Fall)**

**Date and place:** Wednesday November 17, 2021, Hjørnet and Zoom, 12:15.

**Speaker:** Jorge Hidalgo Calderon, University of Bergen

**Title:** The Geometric Maximum Principle and the Alexandrov Theorem

**Abstract:**

I will explain the geometric maximum principle and use it to prove one of the milestones in the global theory of constant mean curvature hypersurfaces embedded in the euclidean space: the celebrated Alexandrov Theorem. The technique employed in this proof, known in the literature as Alexandrov reflection method, has proved itself quite useful in several branches of mathematics, particularly in PDE's and in Differential Geometry.

**Date and place:** Wednesday November 10: No seminar

**Date and place:** Wednesday November 3, 2021, Hjørnet and Zoom, 12:15.

**Speaker:** Stein Andreas Bethuelsen, University of Bergen

**Title:** Random walks in dynamic random environment

**Abstract:**

The classical random walk model is a central object within mathematics that e.g. models the propagation of a particle through a medium. In this talk we will focus on certain extensions of the random walk model allowing for random irregularities in the medium. This theory is closely linked to what is known as stochastic homogenization theory.

In the first part of the talk we will review classical results about such random walks in random environment which reveal a rich phenomenology concerning their asymptotic behavior. At the same time we will see that some of the most fundamental questions remain mathematically unsolved.

In the second half we will focus on the case where the environment evolves dynamically with time. In this case, the mixing properties of the dynamics will play an important role, both concerning the phenomenology and the available mathematical theory. This latter part will partly be based on joint work with Florian Völlering (University of Leipzig).

**Date and place:** Wednesday October 27, 2021, Hjørnet and Zoom, 12:15.

**Speaker:** René Langøen, University of Bergen

**Title:** Complex structures on manifolds and holomorphic maps

**Abstract:**

I will explain complex holomorphic manifolds and give the definitions of holomorphic maps on them. We shall define the tangent space of a complex holomorphic manifold at a point p, as the vector space of C-linear derivations of holomorphic function germs at p. However, it is not immediately clear how to describe this vector space in more detail. We will thus follow a different approach, taking the real 2n dimensional vector space obtained from the smooth structure on the manifold, impose an almost complex structure on it and then complexify it. We will obtain several linear (real and complex) isomorphisms relating the different constructions.

**Date and place:** Wednesday October 20, 2021, Hjørnet and Zoom, 12:15.

**Speaker:** Jean-Claude Saut, Université Paris-Saclay

**Title:** Old and new on the intermediate long wave equation

**Abstract:**

The Intermediate Long Wave equation (ILW) is a classical asymptotic weakly nonlinear model of internal waves in stratified fluids. It also turns out to be completely integrable.We will first recall the rigorous derivation of the ILW and related models and survey the already known results on the Cauchy problem, mainly obtained by "PDE" techniques. On the other hand there is so far norigorous results on the Cauchy problem, even for small initial data, using Inverse Scatering techniques. In particular the soliton resolution, clearly shown by numerical simulations, is not yet proven.We will then present new results on the qualitative properties of solutions obtained in a joint work with Claudio Munoz and Gustavo Ponce.

**Date and place:** Wednesday October 13, 2021, Hjørnet and Zoom, 12:15.

Contact the seminar organizer for link.

**Speaker:** Irina Markina, University of Bergen

**Title:** H-type Lie algebras

**Abstract:**

I will describe 2-step nilpotent Lie algebras, closely related to the Clifford algebras. The H(eisenberg)-type Lie algebras, introduced by Aroldo Kaplan at 1980 for the study of hypoelliptic operators. I will try to explain the relation of H-type Lie algebras to the Clifford algebras and maybe to the composition of quadratic forms. For that, I will briefly revise the definition of Clifford algebras and their representations. I will introduce modules of Clifford algebras versus admissible modules.If time allows I will sketch such questions as the existence of rational structure constants, classification up to isomorphism, Atiyah-Bott periodicity inherited from the Clifford algebras.

**Date and place:** Wednesday October 6: No seminar

**Date and place:** Wednesday September 29, 2021, Hjørnet and Zoom, 12:15.

Contact the seminar organizer for link.

**Speaker:** Gianmarco Vega-Molino, University of Bergen

**Title:** H-type Foliations

**Abstract:**

We discuss H-type foliations, a topic jointly introduced with Fabrice Baudoin, Erlend Grong, and Luca Rizzi. Arising as the sub-Riemannian geometries transverse to Riemannian foliations they are ideally suited to study at the intersection of "extrinsic" and "intrinsic" approaches to sub-Riemannian geometry. We begin with a broad introduction to sub-Riemannian geometry suitable to non-experts and will cover several topics, including forthcoming work on the holonomy of these spaces.

**Date and place:** Wednesday September 22, 2021, Hjørnet and Zoom, 12:15.

Contact the seminar organizer for link.

**Speaker:** Erlend Grong, University of Bergen

**Title:** Statistics of manifolds and most probable paths

**Abstract:**

For a collection of data on a nonlinear space, we cannot use pluss to discuss things such as mean and covariance. In our seminar, we will discuss how to use stochastic processes to model normal distribiutions with a given covariance on a Riemannian manifold. For actual computation, a good tool to use are most probable path; the path a random time dependent variable is most likely to follow. These can be descibed using sub-Riemannian geometry. We will give some explicit formulas for such curves and applications. These results are from a joint work with Stefan Sommer.

**Date and place:** Wednesday September 15: No seminar.

**Date and place:** Wednesday September 8, 2021, Zoom, 15:15. Contact the seminar organizer for link.

**Speaker:** Vitali Vougalter

**Title:** On the solvability of some systems of integro-differential equations with anomalous diffusion in higher dimensions

**Abstract:**

The work deals with the studies of the existence of solutions of a system of integro-differential equations in the case of theanomalous diffusion with the negative Laplace operator in a fractional power in R^d, d=4,5. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

**Date and place:** Wednesday September 1, 2021, Delta + Lunchroom and Zoom, 12:15. Contact the seminar organizer for link.Joint session with the Algebra seminar.

**Speaker:** Adrien Laurent

**Title:** Exotic aromatic B-series and multiscale methods for the integration of ergodic and stiff stochastic dynamics in R^d or on manifolds

**Abstract:**

After a brief summary of the standard results for the weak and the long-time numerical integration of stochastic problems, we propose a new multirevolution method for the weak integration of SDEs with a fast stochastic oscillation. Then, we present a new formalism of Butcher-series, called exotic aromatic B-series, for creating high order integrators for sampling the invariant measure of ergodic stochastic differential equations in R^d and on manifolds. In the particular case of overdamped Langevin dynamics, we obtain the order conditions for a class of Runge-Kutta methods, extend the results to postprocessors and partitioned problems in the context of R^d, and introduce an integrator of order two in the manifold case. We conclude with a new method whose accuracy remains the same in R^d and on a manifold for the integration of penalized Langevin dynamics.

Of course, depending on the knowledge of the audience and on the interactions during the talk, I can adapt my presentation.

**2021 (Spring)**

**Date and place:** May 10, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

**Speaker:** Hans Z. Munthe-Kaas, University of Bergen

**Title:** Lie–Butcher series for geodesic flows

**Abstract:**

We introduce series developments similar to Butcher’s B-series for geodesic flows on manifolds equipped with a general affine connection. This includes the Levi–Civita connection on Riemannian metric spaces as a special case. This novel theory paths new ways for analysing numerical integration schemes based on geodesic flows as well as the study of rough paths in affine geometries.

(joint work with Kurusch Ebrahimi-Fard and Dominique Manchon)

**Date and place:** May 3, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

**Speaker:** Eirik Berge, NTNU Trondheim

**Title:** The Feichtinger Algebra - Too Good to be True?

**Abstract:**

In this talk, I will motivate and explain the Feichtinger algebra. The algebra was invented in the '80s by Hans Georg Feichtinger and has, since the turn of the century, been a central player in time-frequency analysis. Interestingly, the Feichtinger algebra can be defined in a multitude of ways, emphasizing e.g. representation theory, geometric decompositions, or classical analysis. If time permits, I will talk briefly about my own research in the area towards the end of the talk. The goal of the talk is to convince you that the Feichtinger algebra is a beautiful piece of modern mathematics that should be more well-known. I've tried to make the talk accessible for master students.

**Date and place:** April 19, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

**Speaker:** Frédéric Valet, University of Bergen

**Title:** Growth of Sobolev norms for solutions of the Zakharov-Kuznetsov equation in 2D

**Abstract:**

This is a joint work with Raphaël Côte. For a non dispersive equation like the wave equation, a wave packet moves at a fixed velocity. For a dispersive equation like the Zakharov-Kuznetsov equation (ZK) in 2D, a wave packet can be influenced by the dispersive effects, which means that the energy moves from a low wavenumber to a higher wavenumber along the time. This displacement of frequencies is known in physics like the energy cascade phenomenon. In other words, the influence of the dispersive effect is determined by the evolution of Sobolev norms along the time. In this talk, I will detail the cascade phenomenon, explain how to obtain first an exponential bound of the growth of Sobolev norms and how to improve it into a polynomial bound.

**Date and place:** March 15, 2021, Auditorium 3 and Zoom, 14:15.

Contact the seminar organizer for link.

**Speaker:** Erlend Grong, University of Bergen

**Title:** Geometry of sub-Riemannian (2,3,5) manifolds.

**Abstract:**

We will give details of finding a canonical choice of grading and connection on a sub-Riemannian manifold with growth vector (2,3,5).

The first part of the talk will be to explain what the previous words mean. Then we will give our result and discuss how it relates to the sub-Riemannian equivalence problem. In particular, we will give a flatness theorem.

The talk focuses on a special case of results found in

https://arxiv.org/abs/2010.05366

**2020**

Date | Speaker | Institution | Title |

21/01/2020 | Zhenyu Wang | UiB & Harbin Institute of Technology | Numerical simulations for stochastic differential equations on manifolds by stochastic symmetric projection method |

28/01/2020 | Erlend Grong | UiB | Functions of random variable, inequalities on path space and geometry |

11/02/2020 | Torstein Nilssen | UiA | Introduction to rough paths. Introductory part to workshop "Young researchers between geometry and stochastic analysis". |

18/02/2020 | Sylvie Vega-Molino | University of Connecticut | Comparison Theorems on H-type Foliations, an Invitation to sub-Riemannian Geometry |

18/02/2020 | Jacek Jendrej | University Paris 13 | Strongly interacting kink-antikink pairs for scalar fields on a line |

25/02/2020 | Frédéric Vallet | Université de Strasbourg | On the multi-solitons of the Zakharov-Kuznetsov equations |

10/03/2020 | Adan Corcho | Federal University of Rio de Janeiro and Miguel Alejo, University of Cordoba | Stability of nonlinear patterns in low dimensional Bose gases |

01/09/2020 | Irina Markina | UiB | One parametric family of geodesics on the Stiefel manifold |

08/09/2020 | Didier Pilod | UiB | On the unique continuation of solutions to nonlocal non-linear dispersive equations |

22/09/2020 | Francesca Tripaldi | University of Bern | Rumin complex on nilpotent Lie groups and applications |

29/09/2020 | Erlend Grong | UiB | On the equivalence problem on sub-Riemannian manifolds |

06/10/2020 | Stefan Sommer | University of Copenhagen | Stochastic Shape Analysis and Probabilistic Geometric Statistics |

20/10/2020 | Mario Maurelli | Università degli Studi di Milano | Regularization by noise |

03/11/2020 | Razvan Mosincat | UiB | Global well-posedness and scattering for the Dysthe equation in L^2(R^2) |

**Date and place:** November 3, 2020, Auditorium 3 and on Zoom

**Speaker:** Razvan Mosincat, University of Bergen

**Title:** Global well-posedness and scattering for the Dysthe equation in L^2(R^2)

**Abstract:** The Dysthe equation arises, for example, as an asymptoticmodel of the water waves system in the so-called modulation regime. In the first part we will recall the context in which this equation wasfirst derived by Kristian B. Dysthe. We will also review priorwell-posedness results of the initial-value problem.

In the second part we employ a change of variables, sharpStrichartz and bilinear estimates on the linear evolution, and the Fourier restriction norm method in the framework of U^2-, V^2-spaces to prove the small data global well-posedness and scattering of the Dysthe equation in the critical space L^2(R^2).

This result is a recent joint work with Didier Pilod and Jean-Claude Saut.

**Date and place:** October 20, 2020, Zoom, 14:15.

**Speaker:** Mario Maurelli, Università degli Studi di Milano (UniMi

**Title:** Regularization by noise

**Abstract:** We say that a regularization by noise phenomenon occurs if a possibly ill-posed ordinary or partial differential equation becomes well- posed by adding a suitable noise term. This phenomenon is counter-intuitive at first (one adds an irregular noise term to an irregular deterministic part and gets well-posedness). Nevertheless it has been shown for a wide class of ODEs and some PDEs, and has attracted a lot of attention in recent years, with the long-term goal of proving regularization by noise for equations coming from physics, especially fluid dynamics (see e.g. [Flandoli, St. Flour Lect. Notes, 2015]).

In the first part of the talk, I will review some results and techniques of regularization by noise for ordinary differential equations.

In the second part of the talk, I will review regularization by noise for linear PDEs of transport-type and I will show a regularization by noise result for a scalar conservation law with space-irregular drift (the latter is a joint work with Benjamin Gess).

**Date and place:** October 6, 2020, Zoom

**Speaker:** Stefan Sommer, University of Copenhagen

**Title:** Stochastic Shape Analysis and Probabilistic Geometric Statistics

**Abstract:** Analysis and statistics of shape variation can be formulated in a geometric setting with geodesics modelling transitions between shapes. In the talk, I will show how such smooth geodesic models can be extended to account for noise resulting in stochastic shape evolutions and stochastic shape matching algorithms. I will connect these ideas to geometric statistics, the statistical analysis of general manifold valued data. Taking a probabilistic approach to geometric statistics leads to a geometric version of principal component analysis, and most probable paths for the resulting stochastic flows can be identified as geodesics for a sub-Riemannian metric on the frame bundle of the underlying manifold.

**Date and place:** September 29, 2020, 14:15-16, Zoom

**Speaker:** Erlend Grong, University of Bergen

**Title:** On the equivalence problem on sub-Riemannian manifolds

**Abstract:** How to determine if two objects are "the same"? Every topic of mathematics has their own notion of equivalence, usually refering to the existance of a certain map preserving all of the properties we are interested in. Isomorphisms in algebra, homeomorphisms in topology and isometries in differential geometry. However, given two objects from all of these examples, it is usually not evident whether or not such an "equivalence map" exists

We will concern ourselves with this problem in differential geometry. In the first hour, we will give an introduction and discuss the problem in Riemannian geometry. We will in particular look at the role curvature plays in this question. For the second hour, we will look at Cartan geometry and its applications to sub-Riemannian manifolds.

**Date and place:** September 22, 2020, 14:15-15:00, Zoom.

**Speaker:** Francesca Tripaldi, University of Bern

**Title:** Rumin complex on nilpotent Lie groups and applications

**Abstract: **The present work focuses on introducing the tools needed to extend the construction of the Rumin complex to arbitrary nilpotent Lie groups (not necessarily gradable ones). This then enables the direct application of non-vanishing results for the $\ell^{q,p}$ cohomology to all nilpotent Lie groups.

**Date and place:** September 8, 2020, 14:15-16:00, Auditorium 3

**Speaker:** Didier Pilod, University of Bergen**Title:** On the unique continuation of solutions to nonlocal non-linear dispersive equations

**Abstract: **The first part of this talk is an introduction to the unique continuation problem in PDE. We will focus on elliptic problem and explain how to deal with nonlocal equations through the Caffarelli-Silvestre extension.In the second part, we explain how these ideas apply to a large class of nonlocal dispersive equations. If time allows, we will also discuss unique continuation properties for the water waves equations.This talk is based on a joint work with Carlos Kenig (Chicago), Gustavo Ponce (Santa Barbara) and Luis Vega (Bilbao).

**Date and place:** September 1, 2020, 14:15-16:00, Auditorium 3

**Speaker:** Irina Markina, University of Bergen

**Title:** One parametric family of geodesics on the Stiefel manifold

**Abstract:** We start from a very mild introduction to the family of orthogonal and skew-symmetric matrices. We introduce a one-parametric family of metrics on the direct product of orthogonal matrices. Then we explain what is the Stiefel manifold and how it is related to the group of orthogonal matrices. The final goal is to explain how geodesics on the Stiefel manifold can be found by making use of the geodesics on the group of orthogonal matrices. The constructed family of geodesics for the introduced one parametric family of metrics includes various known cases used in the applied mathematics.This is a joint work with K. Hueper (University of Wurzburg) and F. Silva Leite (University of Coimbra).

**Date and place:** March 10, 2020, Seminar room Sigma

**Speaker:** Adan Corcho, Federal University of Rio de Janeiro and Miguel Alejo, University of Cordoba

**Title:** Stability of nonlinear patterns in low dimensional Bose gases

**Abstract:** In this talk we will present recent results on the study of theorbital stability properties of the simplest nonlinear pattern in lowdimensional Bose gases, the black soliton solution.In the first part of the talk, we will introduce basic notions and conceptsrelated with this quantum model as well as physical and mathematicalmotivations to approach that problem.In the second part of the talk, we will present a more detailed scheme ofthe main result of this work on stability of the black soliton. This is asolution of a one dimensional nonintegrable defocusing Schrödinger model,represented by the quintic Gross-Pitaevskii equation (5GP). Once the blacksoliton is characterized as a critical point of the associated Ginzburg-Landau energy of the 5GP, I will show some coercivity propertiesof that energy around the black (and dark) soliton. We will also explain howto impose suitable orthogonality conditions and how to control the growthof some modulation parameters to finally prove that perturbations generatedby the symmetries of the 5GP stay close to the black soliton in the energyspace.

**Date and place:** February 25, 2020, Seminar room Sigma.

**Speaker:** Frédéric Vallet, Université de Strasbourg

**Title**: On the multi-solitons of the Zakharov-Kuznetsov equations.

**Abstract:** In the field of dispersive equations, traveling waves are one of the most fundamental objects. Those waves, also called solitons, keep their velocity and form along time, and are considered as elementary bricks of dispersive equations. The soliton resolution conjecture states that in long time, a solution of Zakharov-Kuznetsov equations (ZK) can be decomposed into a sum of solitons plus a small remainder. In the first talk, I will introduce the equations (ZK) and the context of those equations, then substantiate the existence and properties of solitons, and conclude with the existence and the uniqueness of solutions behaving in long time as a sum of decoupled solitons: the multi-solitons. The second talk will be dedicated to prove the construction of multi-solitons.

**Date and place:** February 18, 2020, at 14:15, Seminar room Sigma

**Speaker:** Jacek Jendrej, University Paris 13

**Title**: Strongly interacting kink-antikink pairs for scalar fields on a line.

**Abstract:** I will present a recent joint work with Michał Kowalczyk and Andrew Lawrie. A nonlinear wave equation with a double-well potential in 1+1 dimension admits stationary solutions called kinks and antikinks, which are minimal energy solutions connecting the two minima of the potential. We study solutions whose energy is equal to twice the energy of a kink, which is the threshold energy for a formation of a kink-antikink pair. We prove that, up to translations in space and time, there is exactly one kink-antikink pair having this threshold energy. I will explain the main ingredients of the proof.

**Date and place:** February 18, 2020, 15:15. Seminar room Sigma

**Speaker:** Gianmarco Molino, University of Connecticut

**Title**: Comparison Theorems on H-type Foliations, an Invitation to sub-Riemannian Geometry.

**Abstract: **Sub-Riemannian geometry is a generalization of Riemannian geometry to spaces that have a notion of distance, but have restrictions on the valid directions of motion. These arise in a natural way in remarkably many settings.This talk will include a review of Riemannian geometry and an introduction to sub-Riemannian geometry. We'll then introduce the notion of H-type foliations; these are a family of sub-Riemannian manifolds that generalize both the K-contact structures arising in contact geometry and the H-type group structures. Our main focus will be recent results giving uniform comparison theorems for the Hessian and Laplacian on a family of Riemannian metrics converging to sub-Riemannian ones. From this we can conclude a sharp sub-Riemannian Bonnet-Myers type theorem.

**Date and place:** February 11, 2020, Seminar room Sigma

**Speaker:** Torstein Nilssen, University of Agder

**Title**: Introduction to rough paths. Introductory part to workshop "Young researchers between geometry and stochastic analysis".

**Date and place:** January 28, 2020, Seminar room Delta

**Speaker:** Erlend Grong, University of Bergen

**Title**: Functions of random variable, inequalities on path space and geometry.

**Abstract: **We will give a quick introduction of functions with random inputs as functions on path space.We describe how to develop a functional analysis of such functions, first over flat space and then over curves space.We will end by describing the relationship between bounded curvature and functional inequalities on path space.We will end with presenting some new results relating functional inequalities on path space and curvature of sub-Riemannian spaces.The results are obtained in collaboration with Li-Juan Cheng and Anton Thalmaier (arXiv:1912.03575).

**Date and place:** January 21, 2020, Seminar room Delta

**Speaker:** Zhenyu Wang, University of Bergen and Harbin Institute of Technology at Weiha

**Title**: Numerical simulations for stochastic differential equations on manifolds by stochastic symmetric projection method.

**Abstract:** Stochastic standard projection technique, as an eﬃcient approach to simulate stochastic diﬀerential equations on manifolds, is widely used in practical applications. However, stochastic standard projection methods usually destroy the geometric properties (such as symplecticity or reversibility), even though the underlying methods are symplectic or symmetric, which seriously aﬀect long-time behavior of the numerical solutions. In this talk, a modiﬁcation of stochastic standard projection methods for stochastic diﬀerential equations on manifolds is presented. The modiﬁed methods, called the stochastic symmetric projection methods, remain the symmetry and the ρ -reversibility of the underlying methods and maintain the numerical solutions on the correct manifolds. The mean square convergence order of these methods are proved to be the same as the underlying methods’. Numerical experiments are implemented to verify the theoretical results and show the superiority of the stochastic symmetric projection methods over the stochastic standard projection methods.

**2019**

Date | Speaker | Institution | Title |

22/01 | Arnaud Eychenne | UiB | On the stability of 2D dipolar Bose-Einstein condensates |

29/01 | Irina Markina | UiB | On normal and abnormal geodesics on the sub-Riemannian geometry |

05/02 | Mauricio Godoy Molina | University De La Frontera | The Volterra equation on manifolds |

12/02 | Dider Pilod | UiB | ?? |

19/02 | Eirik Berge | NTNU | Decomposition Spaces From a Metric Geometry Standpoint |

26/02 | Erlend Grong | UiB | ?? |

02/04 | Eivind Schneider | UiT | ?? |

30/04 | Razvan Mosincat | UiB | ?? |

07/05 | Claudio Munos | University of Chile | ?? |

27/08 | Adán J. Corcho | Universidade Federal do Rio de Janeiro | On the global dynamics for some dispersive systems in nonlinear optics |

03/09 | Alexander Schmeding | UiB | An invitation to infinite dimensional geometry |

10/09 | Razvan Mosincat | UiB | Unconditional uniqueness of solutions to the Benjamin-Ono equation |

17/09 | Luc Molinet | Université de Tours | On the asymptotic stability of the Camassa-Holm peakons |

01/10 | Pavel Gumenyuk | UiS | Univalent functions with quasiconformal extensions |

14/10 | Alexander Schmeding | UiB | Crash course in Brownian motion and stochastic integration, Part I |

22/10 | Alexander Schmeding | UiB | Crash course in Brownian motion and stochastic integration, Part II |

29/10 | Niels Martin Møller | University of Copenhagen | Mean curvature flow and Liouville-type theorems |

29/10 | Razvan Monsincat | UiB | Crash course in Brownian motion and stochastic integration, Part III |

06/11 | Razvan Monsincat | UiB | Crash course in Brownian motion and stochastic integration, Part IV |

12/11 | Evgueni Dinvay | UiB | The Whitham solitary waves |

12/11 | Achenef Temesgen | UiB | Dispersive estimates for the fractal wave equation |

14/11 | Erlend Grong | UiB | Crash course in Brownian motion and stochastic integration, Part V |

16/11 | Erlend Grong | UiB | Crash course in Brownian motion and stochastic integration, Part VI |

26/11 | Jonatan Stava | UiB | Cartan Connection in Sub-Riemannian Geometry |

**Date and place:** November 26, 2019, Seminar room Sigma

**Speaker:** Jonatan Stava, University of Bergen**Title**: Cartan Connection in Sub-Riemannian Geometry.**Abstract:**

If we can associate a Cartan geometry with a sub-Riemannian manifold, the Cartan connection will give a notion of curvature. In the seminar we will look at how we can associate a Lie algebra to each point of a bracket generating sub-Riemannian manifold which is called the sub-Riemannian symbol of the manifold. In a paper by T. Morimoto (2006), he describes how one can obtain a Cartan geometry from a sub-Riemannian manifold with constant symbol in a canonical way. We will see how this method apply to sub-Riemannian manifolds with the Heisenberg Lie algebra as constant symbol.

**Date and place:** November 16, 2019, Seminar room Sigma

**Speaker:** Erlend Grong, University of Bergen**Title**: Crash course in Brownian motion and stochastic integration, Part VI

**Date and place:** November 12, 2019, Seminar room Sigma

**Speaker:** Achenef Temesgen, University of Bergen**Title**: Dispersive estimates for the fractal wave equation

**Date and place:** November 14, 2019, Seminar room Sigma

**Speaker:** Erlend Grong, University of Bergen**Title**: Crash course in Brownian motion and stochastic integration, Part V

**Date and place:** November 12, 2019, Seminar room Sigma

**Speaker:** Evgueni Dinvay, University of Bergen**Title**: The Whitham solitary waves.**Abstract:**

The Whitham equation was proposed as an alternative to the Korteweg-de Vries equation. Having the same nonlinearity as the latter, it featuresthe same linear dispersion relation as the full water-wave problem. It is known to be locally well-posed and admitting wave breaking.It was also proved to posses solitary wave solutions, firstly, in 2012 by Ehrnstrom, Groves and Wahlen.They have used a variational approach, reformulating the problem as a constrained minimization problem.To extract a converging minimizing sequence they have appealed to the Concentration Compactness principal.An alternative elegant proof was given by Stefanov and Wright recently in 2018. They have rescaled the Whitham travelling wave equationintroducing a small parameter that lead in the limit to the KdV travelling wave equation.Existence comes from appeal to the implicit function theorem. In the talk we will mostly discuss their approach in more details.

**Date and place:** November 6, 2019, Seminar room Sigma

**Speaker:** Razvan Monsincat, University of Bergen**Title**: Crash course in Brownian motion and stochastic integration, Part IV

**Date and place:** October 29, 2019, Seminar room Sigma

**Speaker:** Razvan Monsincat, University of Bergen**Title**: Crash course in Brownian motion and stochastic integration, Part III

**Date and place:** October 29, 2019, Seminar room Sigma

**Speaker:** Niels Martin Møller, University of Copenhagen**Title**: Mean curvature flow and Liouville-type theorems**Abstract:**

In the first part we review the basics of mean curvature flow and its important solitons, which are model singularities for the flow, with a view towards minimal surface theory and elliptic PDEs. These solitons have been studied since the first examples were found by Mullins in 1956, and one may consider the more general class of ancient flows, which arise as singularity models by blow-up. Insight from gluing constructions indicate that classifying them as such is not viable, except e.g. under various curvature assumptions.In the talk's second part, however, without restrictions on curvature, we will show that if one applies certain "forgetful" operations - discard the time coordinate and take the convex hull - then there are only four types of behavior. To show this, we prove a natural new "wedge theorem" for proper ancient flows, which adds to a long story: It is reminiscent of a Liouville theorem (as for holomorphic functions), and generalizes our own wedge theorem for self-translaters from 2018 (a main motivating example throughout the talk) that implies the minimal surface case by Hoffman-Meeks (1990) which in turn contains the classical theorems by Omori (1967) and Nitsche (1965).This is joint work with Francesco Chini (U Copenhagen).

**Date and place:** October 22, 2019, Seminar room Sigma

**Speaker:** Alexander Schmeding**Title**: Crash course in Brownian motion and stochastic integration, Part II

**Date and place:** October 14, 2019, Seminar room Sigma

**Speaker:** Alexander Schmeding**Title**: Crash course in Brownian motion and stochastic integration, Part I

**Date and place:** October 1, 2019, Seminar room Sigma

**Speaker:** Pavel Gumenyuk, University of Stavanger**Title**: Univalent functions with quasiconformal extensions**Abstract:**

Univalent functions (i.e. conformal mappings) admitting quasiconformal extensions is a classical topic in Geometric Function Theory, closely related Teichmüller Theory. We consider the class S(k), 0<k<1, of all univalent functions in the unit disk (suitably normalized at the origin) which are restrictions of k-quasiconformal automorphisms of the complex plane. One of the basic tools for finding sufficient conditions for being an element of S(k) is a construction of quasiconformal extensions based on Loewner’s parametric method, discovered by Jochen Becker in 1972. Becker’s extensions have some special properties not shared by generic quasiconformal mapping; in particular, the corresponding class S^B(k) is a proper subset of S(k). This talk is based on recent joint works with István Prause and with Ikkei Hotta. We give a complete characterization of Becker’s extensions in terms of the Beltrami coefficient. This result puts some light over the relationship between the classes S^B(k) and S(k).Our special interest to the class S^B(k) is due to the fact that it admits a parametric representation. Unfortunately, no similar results are known for the whole class S(k).Sharp estimates of the Taylor coefficients in classes of holomorphic functions is an old problem. For the class S(k), it is open for all coefficients a_n, n>2.R. Kühnau and W. Niske in 1977 raised a question whether there exists k_0>0 such that the minimum of |a_3| in S(k) equals k for all 0<k<k_0.Using Loewner’s parametric representation of S^B(k), we show that such a k_0 does not exists. (This disproves a previously known result in this direction by S. Krushkal.)

**Date and place:** September 17, 2019, Seminar room Sigma

**Speaker:** Luc Molinet, Université de Tours**Title**: On the asymptotic stability of the Camassa-Holm peakons**Abstract:** The Camassa-Holm equation possesses peaked solitary waves called peakons. We prove a rigidity result for uniformly almost localized (up to translations) H^1-global solutions of the Camassa-Holm equation with a momentum density that is a non negative finite measure.More precisely, we show that such solution has to be a peakon. As a consequence, we prove that peakons are asymptotically stable in the class of H^1-functions with a momentum density that is a non negative finite measure.

**Date and place:** September 10, 2019, Seminar room Sigma

**Speaker:** Razvan Mosincat, University of Bergen**Title**: Unconditional uniqueness of solutions to the Benjamin-Ono equation**Abstract:** The Benjamin-Ono equation (BO) arises as a model PDE for the propagation of long one-dimensional waves at the interface of two lay- ers of fluids with different densities. From the analytical point of view, it poses technical difficulties due to its quasilinear character. The global well- posedness in L^2 of BO was first shown by Ionescu and Kenig by using an intricate functional setting. Later, Molinet and Pilod, and more recently Ifrim and Tataru gave different and simpler proofs.

In this talk, we are interested in the unconditional uniqueness of solutions to BO. Namely, for a given initial data we establish that there is only one solution without requiring any auxiliary condition on the solution itself. To this purpose we will use a method based on normal form reductions.

**Date and place:** September 3, 2019, Seminar room Sigma

**Speaker:** Alexander Schmeding, UiB**Title**: An invitation to infinite dimensional geometry**Abstract:** Many objects in differential geometry are intimately linked with infinite dimensional structures. For example, to a manifold one can associate it's diffeomorphism group which turns out to be an infinite dimensional Lie group. It carries geometric information which are of relevance in problems from fluid dynamics. After a short introduction to infinite-dimensional structures, I will discuss some connections between finite and infinite dimensional differential geometry.As a main example we will then consider the Euler equation of an incompressible fluid. Due to an observation by Arnold and the work of Ebin and Marsden, one can reformulate this partial differential equation as an ordinary differential equation, but on an infinite dimensional manifold. Using geometric techniques local wellposedness of the Euler equation can be established. If time permits we will then discuss a stochastic version of these results which is recent work together with M. Maurelli (Milano) and K. Modin (Chalmers, Gothenburg).The talk is supposed to give an introduction to these topics. So we will neither supposes familiarity with infinite-dimensional manifolds and their geometry nor with stochastic analysis.

**Date and place:** August 27, 2019, Seminar room Sigma

**Speaker:** Adán J. Corcho, Universidade Federal do Rio de Janeiro**Title**: On the global dynamics for some dispersive systems in nonlinear optics**Abstract:**

We consider two family of coupled equations in the context of nonlinear optics, whose coupling terms are given by quadratic nonlinearities.

The first system is a perturbation of the classic cubic nonlinear Schrödinger equation by a dissipation delay term induced by the medium (Schrödinger - Debye system). In H^1-critical dimension, we present recent results about an alternative between the possible existence of blow-up solution or the grow of the Sobolev norm with high regularity with respect to the delay parameter of the system. The problem of existence of formation of singularities in finite or infinite time remains open for this system.

The second model is given by the nonlinear coupling of two Schr ̈odinger equations and we will show the formation of singularities in the L^2-critical and super-critical cases using the dynamic coming from the Hamiltonian structure. Furthermore, we derive some stability and instability results concerning the ground state solutions of this model.

**Day:** May 07, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Claudio Munos, Associate Professor, University of Chile, Santiago, Chile

**Day:** April 30, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Razvan Mosincat, Postdoc, UiB

**Day:** April 02, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Eivind Schneider, PhD student, University of Tromsoe

**Day:** February 26, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Erlend Grong, Postdoc, UiB, University of Paris Sud, France

**Day:** February 19, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Eirik Berge, PhD student, Mathematical Department, NTNU

**Title:** Decomposition Spaces From a Metric Geometry Standpoint

**Abstract: **In this talk, I will introduce decomposition (function) spacesand discuss a few concrete examples. These are function spaces which appearin different subgenres of analysis such as harmonic analysis,time-frequency analysis and PDE's. I will explain how one can use metricspace geometry (more precisely large scale geometry) to understand andunify these spaces. Finally, if time, I will discuss how one decompositionspace can (or can not) embed in a geometric way into another decompositionsspace, and how this can be detected by utilizing metric geometry.

**Day:** February 12, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Dider Pilod, BFS Researcher, Mathematical Department, UiB

**Day:** February 05, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Mauricio Godoy Molina, Assistent Professor, University De La Frontera, Temuco, Chile

**Title:** The Volterra equation on manifolds.

**Abstract: **The analysis of integro-differential equations has been envoguefor many years, and many results have been produced by changingslightlythe domain of parameters, changing slightly the space of functions orchanging slightly the notion of derivative. This talk will deal withthelatter, and for reasons that will be discussed at length in the talk.Onthe application side, these equations appear in models of sub-diffusiveprocesses; but for the pure mathematician, if we need to applyconvolutions, we better work in the real line instead of a manifold.In this talk, I will discuss some of the ideas we have been pondering.Theaim is to extend some of the results obtained for Euclidean space toRiemannian manifolds, and to do that we need to fill in many atechnicalanalytic detail. This is a work in progress with Juan Carlos Pozo (University De La Frontera).

**Day:** January 29, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Irina Markina, Professor, Mathematical Department, UiB

**Title:** On normal and abnormal geodesics on the sub-Riemannian geometry

**Abstract: **This is a lecture oriented towards the master students in the groups of Analysis and PDE. We will revise the notion of the Riemannian geodesic, Hamiltonian formalism, and show what kind of new type of geodesics appears in sub-Riemannian geometry.

**Day:** January 22, 2019

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Arnaud Eychenne, PhD student, Mathematical Department, UiB

**Title:** On the stability of 2D dipolar Bose-Einstein condensates

**Abstract: **We study the existence of energy minimizers for a Bose-Einstein condensate with dipole-dipole interactions, tightly confined to a plane. The problem is critical in that the kinetic energy and the (partially attractive) interaction energy behave the same under mass-preserving scalings of the wave-function. We obtain a sharp criterion for the existence of ground states, involving the optimal constant of a certain generalized Gagliardo-Nirenberg inequality.

**2018**

Date | Speaker | Institution | Title |

30/01/2018 | Pilod Didier | UiB | A survey on the generalized KdV equations |

06/12/2018 | Pilod Didier | UiB | Construction of a minimal mass blow up solution of the modified Benjamin-Ono equation |

13/02/2018 | Vincent Teyekpiti | UiB | Riemann Problem for a Hyperbolic System With Vanishing Buoyancy |

27/02/2018 | Henrik Kalisch | UiB | Existence and uniqueness of singular solutions to a conservation law arising in magnetohydrodynamics |

13/03/2018 | Irina Markina | UiB | Geodesic equations in sub-Riemannian geometry |

20/03/2018 | Matteo Rafaelli | Danmark Technical University | Flat approximations of surfaces along curves |

10/04/2018 | Yevhen Sevostianov | Zhytomyr Ivan Franko State University | Geometric Approach in the Theory of Spatial Mappings |

24/04/2018 | Miguel Alejo | Federal University of Santa Catarina | On the stability properties of some breather solutions |

08/05/2018 | Eirik Berge | UiB | Sub-Riemannian Model Spaces of Step and Rank Three |

05/06/2018 | Jean-Claude Saut | Universite Paris Saclay | Existence and properties of solitary waves for some two-layer systems |

28/08/2018 | Didier Pilod | UiB | On the local well-posedness for a full dispersion Boussinesq system with surface tension |

04/09/2018 | Victor Gichev | Sobolev Institute of Mathematics | On intersections of nodal sets |

11/09/2018 | Irina Markina | UiB | Heisenberg group as a subgroup og SU(1,2) |

18/09/2018 | Razvan Mosincat | University of Edinburgh | Low-regularity well-posedness for the derivative nonlinear Schrödinger equation |

25/09/2018 | Sven I. Bokn | UiB | On the elastica problem |

23/10/2018 | Luis Marin | UiB | Geodesics on Generalized Damek-Ricci spaces |

30/10/2018 | Jonatan Stava | UiB | On deRham cohomology |

06/11/2018 | Evgueni Dinvay | UiB | Global well-posedness for the BBM equation |

13/11/2018 | Luca Galimberti | UiO | Well-posedness theory for stochastically forced conservation laws on Riemannain Manifolds |

**Day:** November 13, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Luca Galimberti, Postdoc, University of Oslo

**Title:** Well-posedness theory for stochastically forced conservation laws on Riemannain Manifolds.

**Abstract:** We are given an n-dimensional smooth closed manifold M, endowed with a smooth Riemannian metric h. We study the Cauchy problem for a first-order scalar conservation law with stochastic forcing given by a cylindrical Wiener process W. After providing a reasonable notion of solution, we prove an existence and uniqueness-result for our Cauchy problem, by showing convergence of a suitable parabolic approximation of it. This is achieved thanks to a generalized Ito's formula for weak solutions of a wide class of stochastic partial differential equations on Riemannian manifolds. This is a joint work with K.H. Karlsen (UIO).

**Day:** November 06, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Evgueni Dinvay, PhD student, Mathematical Department, UiB

**Title:** Global well-posedness for the BBM equation.

**Abstract: **The regularized long-wave or BBM equation describesthe unidirectional propagation of long surface water waves.We will regard an initial value problem for the BBM equation.It will be shown how to prove its local well posedness with respect to time in Sobolev spaces H^s on real line applying the fixed point argument.Due to conservation of H^1 norm of solutions we will get automatically global well posedness in H^1.From H^1 the global result will be extended to the case with 0<s<1.

**Day:** October 30, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Jonatan Stava, master student, Mathematical Department, UiB

**Title:** On deRham cohomology

**Abstract: **A smooth introduction to deRham cohomology, undestandable for master students will be given

**Day:** October 23, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Luis Marin, master student, Mathematical Department, UiB

**Title:** Geodesics on Generalized Damek-Ricci spaces.

**Abstract: **Damek-Ricci spaces also called harmonic NA groups are harmonic extensionsof H-type groups and have been studied in great detail in harmonic analysis.We wish to generalize the notion of Damek-Ricci spaces, by loosening therestriction on the metric and allowing it to be not only positive definite.This talk will start by giving the basic definition of a psuedo H-typealgebra and discuss their existence and how they are related to Clifford algebras.Further we define the generalized Damek-Ricci spaces as the semi-direct productof a psuedo H-type group with an abelian group and discuss some propertiesof this space. From here we can furnish this space with a left invariantmetric and consider Damek-Ricci spaces as a Riemannian manifold, were we wantto use Hamiltonian formalism to derive a system of equations, wich soultionsgive us the geodesics on the Damek-Ricci space.

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.**Day:** September 25, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Sven I. Bokn, master student, Mathematical Department, UiB

**Title:** On the elastica problem

**Abstract: **"The problem of *elastica* was first proposed, and partially solved, byJames Bernoulli in the late 1600. The complete solution was attributedEuler in the mid 1700 for his detailed description. The solution set is afamily of curves that appear in many natural phenomena. Loosely speaking,the problem of *elastica* is to find a curve of fixed length and boundaryconditions that has minimal curvature.

In this talk we will derive solutions to the *elastica* problem using basicconcepts from differential geometry and the calculus of variations. If timepermits, we will look at how we might address the problem of *elastica* asan optimal control problem on Lie groups. Furthermore, we will look atsimilarities between this problem and the problem of the rolling sphere."

**Day:** September 18, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Razvan Mosincat, PhD student, University of Edinburgh, UK

**Title:** Low-regularity well-posedness for the derivative nonlinear Schrödinger equation

**Abstract: **Harmonic analysis has played an instrumental role in advancing the study of nonlinear dispersive PDEs such as the nonlinear Schrödinger equation. In this talk, we present a method to prove well-posedness of nonlinear dispersive PDEs which avoids a heavy harmonic analytic machinery. As a primary example, we study the Cauchy problem for the derivative nonlinear Schrödinger equation (DNLS) on the real line. We implement an infinite iteration of normal form reductions (namely, integration by parts in time) and reformulate the equation in terms of an infinite series of multilinear terms. This allows us to prove the unconditional uniqueness of solutions to DNLS in an almost end-point space. This is joint work with Haewon Yoon (National Taiwan University).We will also discuss normal form reductions as used in the so-called /I/-method introduced by Colliander, Keel, Staffilani, Takaoka, and Tao. In particular, we consider DNLS on the torus and prove global well-posedness in an end-point space. We also use a coercivity property in the spirit of Guo and Wu to improve the mass-threshold under which the solutions exist globally in time.

**Day:** September 11, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Professor, Irina Markina, Institute of Mathematics, UiB

**Title:** Heisenberg group as a subgroup og SU(1,2)

**Abstract: **We consider the group SU(1,2) of linear transformations in 3 dimensional complex space preserving the metric of the index 1. We associate a unit ball in 2 dimensional complex space with the homogeneous space of SU(1,2) factorised by the isotropy subgroup preserving the origin in 2 dimensional complex space. We describe the Bruhat decomposition of SU(1,2) containing the Heisenberg group. At the end we present the root decomposition of the Lie algebra of the Lie group SU(1,2), where the Heisenberg algebra is naturally arises.

**Day:** September 04, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Professor, Victor Gichev, Sobolev Institute of Mathematics, Omsk, Russia

**Title:** On intersections of nodal sets

**Abstract: ** The nodal set in a Riemannian manifold is the set of zeroes of a Laplace - Beltrami eigenfunction. We will give a sketch of the proof to the following assertion: if the first de Rham cohomologies of the manifold are trivial, then every pair of nodal sets corresponding to the same eigenvalue has a common point. The manifold is assumed to be compact and connected. If it is homogeneous, then it is possible to obtain an additional information on the intersection of nodal sets: a construction for a prescribed finite subset in a nodal set, estimates of their Hausdorff measures, and some other relating results.

**Day:** August 28, 2018

**Time: **14.15-16.00

**Place:** the seminar room 4A5d (\sigma)

**Speaker:** Researcher Didier Jacques Francois Pilod

**Title:** On the local well-posedness for a full dispersion Boussinesq system with surface tension

**Abstract: **We will prove local-in-time well-posedness for a fully dispersive Boussinesq system arising in the context of free surface water waves in two and three spatial dimensions.Those systems can be seen as a weak nonlocal dispersive perturbation of the shallow-water system. Our method of proof relies on energy estimates and a compactness argument. However, due to the lack of symmetry of the nonlinear part, those traditional methods have to be supplementedwith the use of a modified energy in order to close the *a priori* estimates.

This talk is based on a joint work with Henrik Kalisch (University of Bergen)

**Day:** June 05, 2018

**Time: **12.15-14.00

**Place:** the seminar room 4A5d (\sigma)

**Speaker:** Professor Jean-Claude Saut, Universite Paris Saclay, France

**Title:** Existence and properties of solitary waves for some two-layer systems

**Abstract: **We consider different classes of two-layer systems describing the propagation of internal waves, namely the Boussinesq-Full dispersion systemsand the (one-dimensional) two-way versions of the Benjamin-Ono and Intermediate Long Wave equations. After a brief survey on the derivation of asymptotic models for internal waves,we will establish the existence of solitary wave solutions and prove their regularity and decay properties. This is a joint work with Jaime Angulo Pava.

**Day:** Mai 08, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Eirik Berge, master student, Department of Mathematics, UiB

**Title:** Sub-Riemannian Model Spaces of Step and Rank Three

**Abstract: **The development of Riemannian geometry has been highly influencedby certain spaces with maximal symmetry called model spaces. Their ubiquitypresents itself throughout differential geometry from the classicalGaussian map for surfaces to comparison theorems based on volume, theLaplacian, or Jacobi fields. We will in this talk describe a generalizationof the classical model spaces in Riemannian geometry to the sub-Riemanniansetting introduced by Erlend Grong. We will discuss the Riemannian settingfirst to make the presentation (hopefully) accessible to non-experts. Thenwe move towards giving a quick description of the essential concepts neededin sub-Riemannian geometry before turning to the sub-Riemannian modelspaces. Theory regarding Carnot groups and tangent cones will be used toinvoke a powerful invariant of sub-Riemannian model spaces. These toolswill be used to study the classification of sub-Riemannian model spaces.Finally, we will restrict our focus to model spaces with step and rankequal to three and provide their complete classification.

**Day:** April 24, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Miguel Alejo, Federal University of Santa Catarina, Department of Mathematics, Florianopolis-Santa Catarina, Brasil

**Title:** On the stability properties of some breather solutions

**Abstract: **Breathers are localized vibrational wave packets that appear innonlinear systems, that is almost any physical system, when theperturbations are large enough for the linear approximation to bevalid. To be observed in a physical system, breathers should be stable. Inthis talk, there will be presented some results about the stabilityproperties of breather solutions of different continuous models drivenby nonlinear PDEs.It will be shown how to characterize variationally the breathersolutions of some nonlinear PDEs both in the line and in periodicsettings.Two specific variational characterizations will be analyzed:a) the mKdV equation, it model waves in shallow water, and theevolution of closed curves and vortex patches)b) the sine-Gordon equation: it describes phenomena in particlephysics, gravitation, materials and many other systems.Finally, it will be explain how to prove that breather solutions ofthe Gardner equation are also stable in the Sobolev H^2

**Day:** April 10, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Yevhen Sevostianov, Professor, Zhytomyr Ivan Franko State University, Ukraine

**Title:** Geometric Approach in the Theory of Spatial Mappings

**Abstract: **

Space mappings with unbounded characteristics of quasiconformality havebeen investigated. In particular, we mean the so-called mappings withfinite distortion which are intensively investigated by leadingmathematicians in the last decade. The series of properties of theso-called Q-mappings and ring Q-mappings are obtained. The above mappingsare subtype of the mappings with finite distortion and include the mappingswith bounded distortion by Reshetnyak. In particular, the properties ofdifferentiability and ACL, the analogues of the theorems ofCasoratti-Sokhotski–Weierstrass, Liouville, Picard, Iversen etc. areobtained for the above mappings

**Day:** March 20, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Matteo Rafaelli, PhD student, Danmark Technical University,

**Title:** Flat approximations of surfaces along curves

**Abstract: **Given a (smooth) curve on a surface S isometrically embeddedin Euclidean three-space, we present a method for constructing a flat(i.e., developable) surface H which is tangent to S at all points ofthe curve. In the beginning of the talk we will be revising theclassical concepts of Frenet-Serret frame and Darboux frame, on whichsuch construction is based. We will conclude by briefly discussing how the method generalizes tothe case of Euclidean hypersurfaces.

**Day:** March 13, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Irina Markina, Professor, Department of Mathematics, UiB

**Title:** Geodesic equations in sub-Riemannian geometry

**Abstract:**

At the beginning of the talk we will revise the notion of Levi-Civita connection and relation between geodesics and curves minimizing distance function on a Riemannian manifold. After short definition of sub-Riemannian manifold we consider example of geodesic equation on the Heisenberg group. If time allows we will discuss some possible ways of generalization of equations that are the first variations.

**Day:** February 27, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Henrik Kalisch, Professor, Department of Mathematics, UiB

**Title:** Existence and uniqueness of singular solutions to a conservation law arising in magnetohydrodynamics.

**Abstract:**

Existence and admissibility of singular delta-shock solutions is discussed for hyperbolic systems of conservation laws, with a focus on systems which do not admit classical Lax-admissible solutions to certain Riemann problems.One such system is the so-called Brio system arising in magnetohydrodynamics.For this system, we introduce a nonlinear change of variables which can be used to define a framework in which any Riemann problem can be solved uniquely using a combination of rarefaction waves, classical shock waves and singular shocks.

**Day:** February 13, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Vincent Teyekpiti , PhD student, Department of Mathematics, UiB

**Title:** Riemann Problem for a Hyperbolic System With Vanishing Buoyancy

**Abstract:**

In this talk, we shall study a triangular system of hyperbolic equations which is derived as a model for internal waves at the interface of a two-fluid system. The focus will be on a shallow-water system for interfacial waves in the case of a neutrally buoyant two-layer fluid system. Such a situation arises in the case of large underwater lakes of compressible liquids such as CO2 in the deep ocean which may happen naturally or may be manmade. Depending on temperature and depth, such deposits may be either stable, unstable or neutrally stable, and in this talk, the neutrally stable case is considered.The motion of long waves at the interface can be described by a shallow-water system which becomes triangular in the neutrally stable case. In this case, the system ceases to be strictly hyperbolic, and the standard theory of hyperbolic conservation laws may not be used to solve the Riemann problem. It will be shown that the Riemann problem can still be solved uniquely. In order to solve the system, the introduction of singular shocks containing Dirac delta distributions travelling with the shock is required and the solutions are characterized in integrated form using Heaviside functions. We shall also characterize the solutions in terms of vanishing viscosity regularization and show that the two solution concepts coincide.

**Day:** February 6, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Pilod Didier, BFS researcher, Department of Mathematics, UiB

**Title:** Construction of a minimal mass blow up solution of the modified Benjamin-Ono equation

**Abstract: **

We construct a minimal mass blow up solution of the modified Benjamin-Ono equation (mBO), which is a classical one dimensional nonlinear dispersive model.

Let a positive Q in H^{1/2}, be the unique ground state solution associated to mBO. We show the existence of a solution S of mBO satisfying |S| = |Q\| in L_2 and some asymptotic relations as times approach 0/ This existence result is analogous to the one obtained by Martel, Merle and Raphael (J. Eur. Math. Soc., 17 (2015)) for the mass critical generalized Korteweg-de Vries equation (gKdV). However, in contrast with the gKdV equation, for which the blow up problem is now well-understood in a neighborhood of the ground state, S is the first example of blow up solution for mBO.

\medskipThe proof involves the construction of a blow up profile, energy estimates as well as refined localization arguments, developed in the context of Benjamin-Ono type equations by Kenig, Martel and Robbiano (Ann. Inst. H. Poincaré, Anal. Non Lin., 28 (2011)).Due to the lack of information on the mBO flow around the ground state, the energy estimates have to be considerably sharpened here.

This talk is based on a joint work with Yvan Martel (Ecole Polytechnique)

**Day:** January 30, 2018

**Place:** the seminar room 4A9f (\delta)

**Speaker:** Pilod Didier, BFS researcher, Department of Mathematics, UiB

**Title:** A survey on the generalized KdV equations

**Abstract:**

At the end of the 19th century, Boussinesq, and Korteweg and de Vries introduced a *partial differential equation *(PDE), today known as the Korteweg-de Vries (KdV) equation, to model the propagation of long waves in shallow water. The KdV equation is a nonlinear dispersive PDE admitting solitary waves solutions, also called *solitons*, playing an important role in *fluid mechanics* as well as in other fields of science, such as *plasma physics*.

In this talk, we will focus on the *generalized KdV equations* (gKdV) to explain the different types of mathematical questions arising in the field of *nonlinear dispersive equations*. We will describe the techniques of modern analysis, from *harmonic analysis* to *spectral theory*, introduced to solve them, and talk about some related open problems. Finally, at the end of the talk, we will introduce the problem of minimal mass blow-up solutions that will be discussed next week with more details.

**2017**

Date | Speaker | Institution | Title |

23/01/2017 | Mauricio Antonio Godoy Molina | University de La Frontera | Harmonic maps between Riemannian manifolds I |

30/01/2017 | Mauricio Antonio Godoy Molina | University de La Frontera | Harmonic maps between Riemannian manifolds II |

07/02/2017 | Erlend Grond | Université Paris Sud and UiB | Model spaces in sub-Riemannian geometry |

14/02/2017 | Irina Markina | UiB | Hypo-elliptic partial differential operators and Hormander theorem |

28/02/2017 | Viktor Gonzalez Aguilera | Technological University of Santa Maria, Valparaiso | Limit points in the Deligne-Mumford moduli space |

07/03/2017 | Achenef Tesfahun Temesgen | UiB | Small data scattering for semi-relativistic equations with Hartree-type nonlinearity |

14/03/2017 | Erlend Grong | Université Paris Sud and UiB | The geometry of second order differential operators |

25/03/2017 | Clara Aldana | University of Luxembourg | Spectral geometry on Surfaces |

21/03/2017 | Nikolay Kuznetsov | Russian Academy of Sciences | Babenko's equation for periodic gravity waves on water of finite depth |

28/03/2017 | Eugenia Malinnikova | NTNU | Frequency of harmonic functions and zero sets of Laplace eigenfunctions on Riemannian manifolds |

04/04/2017 | Wolfram Bauer | Leibniz University of Hanover | The sub-Laplacian on nilpotent Lie groups – heat kernel and spectral zeta function |

18/04/2017 | Eirik Berge | UiB | Invertibility of Fredholm operators in the Calkin algebra |

02/05/2017 | Stine Marie Eik | UiB | Riemannian and sub-Riemannian Lichnerowicz estimates |

09/05/2017 | Bhagyashri Nilesh Ingale | UiB | Extremality of the spiral stretch map and Teichmüller map |

09/05/2017 | Kim-Erling Bolstad-Larssen | UiB | Quadratic forms and its relation to Hurwitz' problem, theRadon-Hurwitz function and Clifford algebras |

29/08/2017 | Irina Markina | UiB | Geodesics on the Stiefel manifold I |

05/09/2017 | Irina Markina | UiB | Geodesics on the Stiefel manifold II |

12/09/2017 | Aroldo Kaplan | CONICET-Argentina, University of Massachusetts | The Basic Holographic Correspondence |

19/09/2017 | Eirik Berge | UiB | Principal Bundles and Their Geometry I |

26/09/2017 | Eirik Berge | UiB | Principal Bundles and Their Geometry II |

03/10/2017 | Erlend Grong | Université Paris Sud and UiB, Laboratoire des Signaux et Systemes (L2S) Supelec, CNRS, Universite Paris-Saclay, UiB | Comparison theorems for the sub-Laplacian |

10/10/2017 | Emanuele Bodon | University of Genova | Separability for Banach Spaces of Continuous Functions |

17/10/2017 | Anja Eidsheim | UiB | Module and extremal length in the plane |

24/10/2017 | Sven I. Bokn | UiB | Rolling of a ball |

31/10/2017 | Jorge Luis Lopez Marin | UiB | Introduction to the Damek-Ricci space I |

07/11/2017 | Jorge Luis Lopez Marin | UiB | Introduction to the Damek-Ricci space II |

14/11/2017 | Stine Marie Berge | NTNU | Frequency of Harmonic functions |

November 14, 2017

**Speaker:** Stine Marie Berge, PhD student, NTNU

**Title:** Frequency of Harmonic functions

**Abstract:** In the talk we will look at the frequency of a harmonic function. When the harmonic functionis a homogeneous harmonic polynomial, then the frequency simplycoincides with the degree of the polynomial. The main goal is to showhow the increasing frequency implies that harmonicfunctions satisfy some kind of doubling property. We will show this byintroducing the concept of log-convexity.

October 31 and November 07, 2017

**Speaker:** Jorge Luis Lopez Marin, master student, Mathematical Department, University of Bergen

**Title:** Introduction to the Damek-Ricci space

**Abstract:** In these two talks, we will introduce the notions of Lie algebras, Lie Groups and Damek-Riccispaces. The first talk will go through preliminaries to Damek-Ricci spaces and the second talkwill deal with the Damek-Ricci spaces. The first talk will start by introducing Lie algebras andlook at examples. From there we introduce Lie groups as smooth manifolds with additionalalgebraic structures and look at examples of such smooth manifolds. Then we look at howthese two mathematical objects are connected, namely by the Lie exponential map. We areparticular interested in Heisenberg type algebras and groups, as they play an important rolein Damek-Ricci spaces and will therefore introduce these also. The second talk will be takingthe notions from the first talk and use them construct Damek-Ricci spaces. We will look attwo different realizations of Damek-Ricci.

October 24, 2017

**Speaker:** Sven I. Bokn, bachelor student, Mathematical Department, University of Bergen

**Title:** Rolling of a ball

**Abstract:** In this talk we will talk about the rolling of a ball over a plane or over another ball. We will introduce the necessary geometric background, such as a notion of a surface, frame, orientation, the group of orientation preserving rotations and its Lie algebra. We will deduce the kinematic equation of the rolling motion without slipping and twisting for both cases.

October 17, 2017

**Speaker:** Anja Eidsheim, master student, Mathematical Department, University of Bergen

**Title:** Module and extremal length in the plane

**Abstract:** This talk aims to give a thorough introduction to the notions of module of a family of curves and extremal length in the plane. Relations between module and extremal length, and other interesting classical results will be mentioned. Starting out in the complex plane, the module of two types of classical canonical domains in the theory of conformal mappings, namely quadrilaterals and ring domains, will be explained. The module of curve families provides a natural transition from the theory of conformal maps in the complex plane to more general environments. As a first step towards the generalizations to module and capacity in Euclidean n-space and even further, the focus in the talk will move from a conformal module in the complex plane to module of a family of curves in R^2. Examples of how to find the extremal metric and calculate the module of curve families in both annulus and distorted annulus domains in R^2 will be shown.As students are the intended audience for this talk, no prior knowledge of the topic will be required.

October 10, 2017

**Speaker:** Emanuele Bodon, Exchange student from the Department of Mathematics, University of Genova, Italy

**Title:** Separability for Banach Spaces of Continuous Functions

**Abstract: **In this talk, we will introduce Banach spaces of continuous functions, i.e.we will consider the Banach space of the continuous functions from acompact topological space to the real or the complex numbers with the supnorm (and, more generally, we will consider the space of bounded continuousfunctions on a not necessarily compact topological space).After introducing some important examples, we will deal with the problem ofunderstanding whether such a Banach space is separable or not.We will start from discussing the problem in the mentioned examples andthen give some sufficient conditions and (under some assumptions on thetopological space) also a characterization; doing this will require tointroduce some classical theorems of analysis and topology(Stone-Weierstrass theorem, Urysohn metrization theorem, partition ofunity).

October 3, 2017

**Speaker:** Erlend Grong, postdoc, Universite Paris Sud, Laboratoire des Signaux et Systemes (L2S) Supelec, CNRS, Universite Paris-Saclay and the Mathematical Department, UiB

**Title:** Comparison theorems for the sub-Laplacian

**Abstract: **One of the main ways of observing curvature in a RIemannian manifold is to look at how the distance between two points changes as we move along geodesics.Namely, the second variation of the distance can be determined by using Jacobi fields, which are themselves controlled by curvature. As a result, we get can get an estimate for the Laplacian of the distance if we have a lower bound for the Ricci curvature. This result is called the Laplacian comparison theorem.Applying the same idea for a sub-elliptic operator and its corresponding distance, has turned out to be difficult. There have been some attempts to define analogues of Jacobi fields, but these definitions lead to difficult computations. Using Riemannian Jacobi fields and approximation argument, we are able to obtain comparison theorem in a wide range of cases, which are sharp in the case of sub-Riemannian Sasakian manifolds.Applications to these results is a Bonnet-Myers theorem and the measure contraction property.These results are from joint work with Baudoin, Kuwada and Thalmaier.

September 26, 2017

**Speaker:** Eirik Berge, master student, Mathematical Department, UiB

**Title:** Principal Bundles and Their Geometry (Part 2)

**Abstract:** We will investigate an additional piece of information one canput on a principal bundle; a connection. We will use the fundamental vectorfield developed last time to obtain equivalent formulations and see howthis will lead us to connection and curvature forms on a principal bundle.Lastly, if time, I will go through the Stiefel manifold and discuss howIrina's talk on horizontal lifts from the Stiefel manifold can be put intothe principal bundle framework.

September 19, 2017

**Speaker:** Eirik Berge, master student, Mathematical Department, UiB

**Title:** Principal Bundles and Their Geometry (Part 1)

**Abstract:** In this talk, we will introduce principal bundles and understanda few key examples. An additional piece of information, namely a principalconnection, will be introduced to "lift" the geometry of the base manifoldto the principal bundle. If time permits, we will discuss curvature andholonomy, and how they are related to the geometry of the base manifoldthrough the frame bundle. The talk will be elementary and will not requiremany prerequisites in differential topology or geometry.

September 12, 2017

**Speaker:** Professor Aroldo Kaplan, CONICET-Argentina University of Massachusetts, Amherst

**Title:** The Basic Holographic Correspondence

**Abstract:** The correspondence between Einstein metrics on an open manifold andconformal structures on some boundary, has become a subject of renewedinterest after Maldacena’s elaboration of it into the AdS-CFTcorrespondence. In this talk we will describe some of the mathematics inthe case of the hyperbolic spaces, which already leads to previouslyunknown solutions to Einstein’s equations. Only basic Riemannian Geometrywill required for most of the talk.

August 29 and September 5, 2017

**Speaker:** Professor Irina Markina, Mathematical Department, UiB

**Title:** Geodesics on the Stiefel manifold.

**Abstract:** Geodesics on the Stiefel manifold can be calculated by different ways. We will show how to do it by making use of the sub-Riemannian geodesics on the orthogonal group. We introduce the orthogonal group, the Steifel manifold as a homogeneous manifold of the orthogonal group. We discuss the sub-Riemannian structure induced by the projection map and prove a theorem stated general form of sub-Riemannian geodesics. I will try to be gentle to the audience and make the exposition accessible for master students.

May 09, 2017

**Speaker:** Kim-Erling Bolstad-Larssen, master student, University of Bergen

**Title:** Quadratic forms and its relation to Hurwitz' problem, theRadon-Hurwitz function and Clifford algebras.

**Abstract:** In the seminar, we would like to explain the relation between several objects. Namely we will reveal how composition of quadratic forms is related to the Hurwitz problem. We also explain how it leads to the orthogonal design and Clifford algebras. The classical Radon-Hurwitz function, related to the Clifford algebras generated by a vector space with positive definite scalar product, was extended By Wolfe to the Clifford algebras generated by a vector space with an arbitrary indefinite non-degenerate scalar product. We present the formula and show the algorithm for its calculation. If time allows we will show also the relation of the above mentioned objects to some special Lie algebras.

May 09, 2017

**Speaker:** Bhagyashri Nilesh Ingale, master student, University of Bergen

**Title:** Extremality of the spiral stretch map and Teichmüller map.

**Abstract:** We would like to explain how extremal function in the class of homeomorphic mappings with finite distortion is related to the Teichmüller theory. We start with an introduction to the theory of quasiconformal maps and the notion of the modulus of a family of curves. We define the spiral stretch map and explain its extremality property in the class of mappings with finite distortion.We show that the spiral stretch map is the Teichmüller map by finding the corresponding quadratic differential.

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.

May 02, 2017

**Speaker:** Stine Marie Eik, master student, University of Bergen

**Title:** Riemannian and sub-Riemannian Lichnerowicz estimates.

**Abstract:** We will begin by recalling some of the definitions from differentialgeometry before defining the Laplace operator on manifolds.

Thereafter wewill state some of the most important properties of the Laplace operator and then move on to(hopefully) proving the Bochner formula and the Lichnerowicz estimate inthe Riemannian case. This gives us a bound on the first eigenvalue of theLaplacian for manifolds with positive Ricci curvature. In the second partof the talk we move to sub-Riemannian geometry, where we will define the(rough) sub-Laplacian and discuss the generalization of the Bochner formulaand Lichnerowicz estimate.

April 18, 2017

**Speaker:** Eirik Berge, master student, University of Bergen

**Title:** Invertibility of Fredholm operators in the Calkin algebra.

**Abstract:** The aim of the talk is to present the connection between compactand Fredholm operators on a Banach space. It will begin with anintroduction to the theory of compact operators. We will discuss theFredholm alternative and how this can be used to solve integral equations.Then we will focus on the invertibility of Fredholm operators in the Calkinalgebra and derive properties of Fredholm operators through their relationto compact operators. Lastly, if time will allow, we will describe the most importantinvariant of Fredholm operators; their index.

April 04, 2017

**Speaker:** Wolfram Bauer, Professor, Analysis Institute, Leibniz University of Hanover

**Title:** The sub-Laplacian on nilpotent Lie groups – heat kernel and spectral zeta function.

**Abstract:** We recall the notion of the subLaplacian on nilpotent Liegroups and their homogeneous spaces by a cocompact lattice (nilmanifolds). In the case of step 2 nilpotent groups differentmethods are known for deriving the heat kernel explicitly. Suchformulas can be used to study the spectral zeta function and heattrace asymptotic of the operator and to extract geometric informationfrom analytic objects. One may as well consider these operators ondifferential forms. We recall a matrix representation of the formLaplacian on the Heisenberg group. In the case of one forms thespectral decomposition and the corresponding heat operator will bederived in the talk by André Hänel.

March 28, 2017

**Speaker:** Eugenia Malinnikova, Professor, Mathematical Department, NTNU

**Title:** Frequency of harmonic functions and zero sets of Laplace eigenfunctions on Riemannian manifolds.

**Abstract:** We will discuss combinatorial approach to the distributions of frequency of harmonic functions and its application to estimates of the area of zero sets of Laplace eigenfunctions in dimensions two and three. The talk is based on a joint work with A. Logunov.

March 21, 2017

**Speaker**: Nikolay Kuznetsov, Researcher, Laboratory for Mathematical Modelling of Wave Phenomena,

Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St. Petersburg**Title**: Babenko's equation for periodic gravity waves on water of finite depth**Abstract**: For the nonlinear two-dimensional problem, describing periodic steady waves on water of finite depth in the absence of surface tension, a single pseudo-differential operator equation (Babenko's equation) is considered. This equation has the same form as the equation for waves

on infinitely deep water; the latter had been proposed by Babenko in 1987 and studied in detail by Buffoni, Dancer and Toland in 2000.

Unlike the equation for deep water involving just the $2 \pi$-periodic Hilbert transform C, the equation to be presented in the talk contains an operator which is the sum of C and a compact operator depending on the depth of water.

March 14, 2017

**Speaker:** Clara Aldana, research fellow, University of Luxembourg, Luxembourg

**Title:** Spectral geometry on Surfaces.

**Abstract:** I start the talk by introducing some basic concepts in spectral geometry. One considers the Laplace operator on a manifold and its spectrum. Two manifolds are isospectral if their Laplace spectrum is the same. The isospectral problem asks whether the Laplace spectrum determines the metric of the manifold: "Can one hear the shape of a drum?"I will mention some of the known results and the classical compactness theorem of isospectral metrics on surfaces proved by B. Osgood, R. Phillips and P. Sarnak. Next, I will define the determinant of the Laplacian and explain how it is an important global spectral invariant.I will explain the problems that appear when one wants to study determinants and isospectrality on open surfaces and surfaces whose metrics are singular. To finish I will present some of my results in this area.

March 14, 2017

**Speaker:** Erlend Grong, Postdoc, Université Paris Sud and University of Bergen.

**Title:** The geometry of second order differential operators

**Abstract:** Let L be a second order partial differential operators. We want to show that such operators are associated with a certain shape.If L is an operator in two variables, this shape is a surface. In general, such shapes are objects called Riemannian manifolds.By studying the geometry of these shapes, we are able to get results for L and its heat operator.The above framework depends on the assumption that L is elliptic, and is well understood in this case.We will end the talk by discussing how we can get geometric results for L even when It is not elliptic.

March 7, 2017

**Speaker**: Achenef Tesfahun Temesgen, postdoc of the Mathematical Department, University of Bergen**Title**: Small data scattering for semi-relativistic equations with Hartree-type nonlinearity**Abstract**: I will talk about well-posedness, and scattering of solutions for semi-relativistic equations with Hartree-type nonlinearity to free waves asymptotically as t->\infty.

To do so I will first talk about the dispersive properties of free waves, and Strichartz estimates for the linear wave equations and Klein-Gordon equations.

February 28, 2017

Joint analysis and PDE, and algebraic geometry seminar

**Speaker**: Viktor Gonzalez Aguilera, Technological University of Santa Maria, Valparaiso, Chile**Title**: Limit points in the Deligne-Mumford moduli space

**Abstract**: Let M_g be the moduli space of smooth curves of genus g defined over the complex numbers and Md_g be the set of stable curves of genus g. A well known result of Deligne and Mumford states that to set Md_g of stable curves of genus g can be endowed with a structure of projective complex variety and contains M_g as a dense open subvariety. The stable curves can be seen also from the point of view of Bers as Riemann surfaces with nodes, thus an element of Md_g can be considered as a stable curve or as a Riemann surface with nodes. The singular locus of M_g (or the branch locus) is stratified by equisymmetric stratas M_g(Γ, s, Φ, G) that when non empty are smooth connected locally closed

algebraic subvarieties of M_g. In this talk we present some of the work of A. Costa, R. Díaz and myself, in order to describe the “limits” points of these stratas in terms of their associated dual graphs. We give some explicit examples and their projective realization as families of stable curves.

February 14, 2017

**Speaker**: Irina, Markina, Professor, University of Bergen**Title**: Hypo-elliptic partial differential operators and Hormander theorem**Abstract**: In the talk we introduce the notion of hypo-ellipticity for partial differential operators. First we discuss the hypo-ellipticity of differential operators with constant coefficients. Then we consider a second order differential operator, generated by arbitrary vector fields defined in a domain of the n-dimentional Euclidean space. In the talk we will start to prove the Hormander theorem stating that if the mentioned vector fields are such that among their commutators always there are n linearly independent ones (probably different in different points of the domain), then the operator is hypo-elliptic.

February 07, 2017

**Speaker**: Erlend Grond, Postdoc, Université Paris Sud and University of Bergen**Title**: Model spaces in sub-Riemannian geometry**Abstract**:The spheres, hyperbolic spaces and euclidean space are important reference spaces for understanding Riemannian geometry in general. They also play an important role in comparison results such as the Laplace comparison theorem and the volume comparison theorem. These reference or model spaces are characterized by constant sectional curvature and by their abundance of symmetries.

In recent years, there have been several attempts to define an analogue of curvature for sub-Riemannian manifolds, based on either their geodesic flow or on related hypoelliptic partial differential operators. However, it has not been clear what the reference spaces should be in this geometry, for which we can test these definitions of curvature.

We want to introduce model space by looking at sub-Riemannian spaces with a `maximal’ group of isometries.

These turn out to have a rich geometric structure, and exhibit many properties not found in their Riemannian analogue.

January 23, 2017 and January 30, 2017

**Speaker**: Mauricio Antonio Godoy Molina, Associate Professor, Department of Mathematics, University de La Frontera, Temuco, Chile**Title**: Harmonic maps between Riemannian manifolds**Abstract**: Given two Riemannian manifolds, a harmonic map between them is,loosely speaking, a smooth map that is a critical point of an ad-hoc energy functional. Intuitively, they are the maps that have least "stretching". With this 'definition', one expects geodesics and minimal surfaces to be

included as examples, and indeed they are.

The aims of these talks are to define what are harmonic maps in Riemannian geometry, why are they interesting to some people and what can we say about them. In particular, we will spend some time filling in analytic and geometric prerequisites to study the question of existence of harmonic maps

within a given homotopy class.

**2016**

Date | Speaker | Institution | Title |

13/01/2016 | Alexey Tochin | UiB | A general approach to Schramm-Löwner Evolution and its coupling to conformal field theory |

26/01/2016 | Mauricio Godoy Molina | Universidad de La Frontera | Tanaka prolongation of pseudo H-type algebras |

02/02/2016 | Santiago Díaz-Madrigal | University of Seville | Fixed Points in Loewner Theory. |

09/02/2016 | Irina Markina | UiB | Caccioppoli sets |

16/02/2016 | Alexander Vasiliev | UiB | Moduli of families of curves on Riemann surfaces and Jenkins-Strebel differentials |

23/02/2016 | Evgueni Dinvay | UiB | Eigenvalue asymptotics for second order operators with discontinuous weight on the unit interval |

20/04/2016 | Eirik Berge | UiB | Frèchet spaces as modeling spaces for diffeomorphism groups |

26/04/2016 | Eric Schippers | University of Manitoba | The rigged moduli space of CFT and quasiconformal Teichmuller theory |

03/05/2016 | Anastasia Frolova | UiB | Quasiconformal mappings |

06/09/2016 | Alexander Vasiliev | UiB | Ribbon graphs and Jenkins-Strebel quadratic differentials I |

13/09/2016 | Igor Trushin | Tohoku University | On inverse scattering on star-shaped and sun-type graphs |

20/09/2016 | Alexander Vasiliev | UiB | Ribbon graphs and Jenkins-Strebel quadratic differentials II |

27/09/2016 | Evgueni Dinvay | UiB | Spectral theorem in functional analysis I |

04/10/2016 | Evgueni Dinvay | UiB | Spectral theorem in functional analysis II |

11/10/2016 | Evgueni Dinvay | UiB | Spectral theorem in functional analysis III |

October 11, 2016

**Speaker**: Evgueni Dinvay, Department of Mathematics, UiB

**Title**: Spectral theorem in functional analysis

**Abstract**: I am going to continue with spectral theory of linear operators acting in Hilbert spaces. This time I am going to formulate and prove the spectral theorem for unitary operators. These lectures should be regarded as an addition to the usual functional analysis course (MAT 311) and directed first of all to master and Ph.D. students.

October 4, 2016

-. . ...- . .-. --. --- -. -. .- --. .. ...- . -.-- --- ..- ..- .--. -. . ...- . .-. --. --- -. -. .- .-.. . - -.-- --- ..- -.. --- .-- -.

**Speaker**: Evgueni Dinvay, Department of Mathematics, UiB

**Title**: Spectral theorem in functional analysis

**Abstract**: I am going to continue with spectral theory of linear operators acting in Hilbert spaces. This time I am introducing the notion of spectral measure space. These lectures should be regarded as an addition to the usual functional analysis course (MAT 311) and directed first of all to master and Ph.D. students.

September 27, 2016

**Speaker**: Evgueni Dinvay, Department of Mathematics, UiB

**Title**: Spectral theorem in functional analysis

**Abstract**: I am going to give some basics of spectral theory of linear operators acting in Hilbert spaces. We regard spectral theorem for unitary operators in particular. These lectures should be regarded as an addition to the usual functional analysis course and directed first of all to master and Ph.D. students.

September 20, 2016

**Speaker**: Alexander Vasiliev, Professor, UiB.

**Title**: Ribbon graphs and Jenkins-Strebel quadratic differentials

**Abstract**: This will be the final part of my previous talk.

September 13, 2016**Speaker**: Professor Igor Trushin (Research Center for Pure and Applied Math., Tohoku University, Japan)**Title**: On inverse scattering on star-shaped and sun-type graphs.**Abstract**: We investigate inverse scattering problem for the Sturm-Liouville (1-D Schrodinger) operator on the graph, consisting of a finite number of half-lines joint with either a circle or a finite number of finite intervals. Uniqueness of reconstruction of potential and reconstruction procedure on the semi-infinite lines are established.This is a joint work with Prof.K.Mochizuki.

September 6, 2016

**Speaker**: Alexander Vasiliev, Dept. Math, UiB**Title**: Ribbon graphs and Jenkins-Strebel quadratic differentials**Abstract**: Ribbon (fat) graphs became a famous tool after Kontsevich used a combinatorial description of the moduli spaces of curves in terms of them, which led him to a proof of the Witten conjecture about intersection numbers of stable classes on the moduli space. We want to give some basics on ribbon graphs and their relation to Jenkins-Strebel quadratic differentials on Riemann surfaces.

May 3rd

**Speaker**: Anastasia Frolova, PhD student, Department of Mathematics, UiB.

**Title**: Quasiconformal mappings.

**Abstract**: Quasiconformal mappings are a natural generalization of conformal mappings and are used in different areas of mathematics. We give give a short introduction to quasiconformal mappings in the plane, discuss their geometric and analytic properties.

April 26th

**Speaker**: Eric Schippers, Department of Mathematics, University of Manitoba, Winnipeg, Canada

**Title**: The rigged moduli space of CFT and quasiconformal Teichmuller theory.

**Abstract**: A central object of two-dimensional conformal field theory is the Friedan/Shenker/Segal/Vafa moduli space of Riemann surfaces with boundary parameterizations. D. Radnell and I showed that this moduli space can be identified with the (infinite-dimensional) Teichmuller space of bordered surfaces up to a discontinuous group action. In this talk I will give an overview of joint results with Radnell and Staubach, in which we apply the correspondence between the moduli spaces to both conformal field theory and Teichmuller theory. We will also discuss the relation with the so-called Weil-Petersson class Teichmuller space.

April 20th

**Speaker**: Eirik Berge, Master student of Mathematical Department, UiB

**Title**: Frèchet spaces as modeling spaces for diffeomorphism groups.

**Abstract**: I will give an introduction to Frèchet spaces with motivation towards studying the diffeomorphism group of a compact manifold. When the notion of a smooth maps between Frèchet spaces is developed, then Frèchet manifolds and Lie-Frèchet groups will be defined. It turns out that the diffeomorphism group can be given a Lie-Frèchet group structure. Time: 15.15-16.00 Place: the same Speaker: Stine Marie Eik, Master student of Mathematical Department, UiB Title: The bad behavior of the exponential map for the diffeomorphism group of the circle. Abstract: I will discuss constructions on diffeomorphism groups of compact manifolds and describe their Lie algebra. Moreover, I shall define the exponential map for the diffeomorphism group of the circle and prove that it is not a local diffeomorphism, in contrast with the finite dimensional case. Hence, one of the most powerful tool for studying Lie groups is significantly weakened when we generalize to the Lie-Frèchet groups.

February 23rd

**Speaker**: PhD student Evgueni Dinvay, Dept. of Math., University of Bergen

**Title**: Eigenvalue asymptotics for second order operators with discontinuous weight on the unit interval.

**Abstract**: We consider a second order differential operator on the unit interval with the Dirichlet type boundary conditions. At the beginning of the presentation we will review the well-known results on the subject. There will be given eigenvalue asymptotic, a trace formula for this operator and will be formulated an inverse spectral problem. Then we discuss what results might be extended for the corresponding operator with discontinuous weight.

February 16th

**Speaker**: Professor Alexander Vasil'ev, Dept. of Math., University of Bergen

**Title**: Moduli of families of curves on Riemann surfaces and Jenkins-Strebel differentials

**Abstract**: We give a comprehensive review of the development of the method of extremal lengths (or their reciprocals called moduli) of families of curves on Riemann surfaces, a basic example of which is a punctured complex plane. The dual is the problem of the extremal partition of the Riemann surface. It turns out that the meeting point of these two problem is achieved by quadratic differentials which provide the extremal functions in the moduli problem, and at the same time, define the extremal partitions.

February 9th

**Speaker**: Professor Irina Markina, Dept. of Math., University of Bergen

**Title**: Caccioppoli sets

**Abstract**: It this seminar we will surprisingly see that the characteristic function of rational numbers is continuous (in some sense) and rather crazy sets (like coast of Norway) still have finite length. The talk is an introductory talk to the theory of sets of finite perimeter or Caccioppoli sets. We will start from the revision of notion of a function of bounded variation of one variable and will see what is a natural generalisation to n-dimensional Euclidean space. Later we will use BV functions to define a perimeter of an arbitrary measurable set. We compare the perimeter to the surface area measure. At the end we will see how Caccioppoli set can be defined in an arbitrary metric space.

February 2nd

**Speaker**: Professor Santiago Díaz-Madrigal, University of Seville, Spain

**Title**: Fixed Points in Loewner Theory.

**Abstract**: Starting from the case of semigroups, we analyze (and compare) fixed points of evolution families as well as critical points of the associated vector fields. A number of examples are also shown to clarify the role of the different conditions assumed in the main theorems.

**Speaker:** Professor Manuel Domingo Contreras Márquez, University of Seville, Spain

**Title**: Integral operators mapping into the space of bounded analytic functions

January 26th

**Speaker**: Mauricio Godoy Molina, Assistant Professor, Universidad de La Frontera, Chile.

**Title**: Tanaka prolongation of pseudo H-type algebras

**Abstract**: The old problem of describing infinitesimal symmetries of distributions still presents many interesting questions. A way of encoding these symmetries for the special situation of graded nilpotent Lie algebras was developed by N. Tanaka in the 70's. This technique consists of extending or "prolonging" the algebra to one containing the original algebra in a natural manner, but no longer nilpotent. When this prolongation is of finite dimension, then the algebra we started with is called "rigid", and otherwise it is said to be of "infinite type". The goal of this talk is to extend a result by Ottazzi and Warhurst in 2011, to show that a certain class of 2-step nilpotent Lie algebras (the pseudo $H$-type algebras) are rigid if and only if their center has dimension greater or equal than three. This is a joint work with B. Kruglikov (Tromsø), I. Markina and A. Vasiliev (Bergen).

January 13th

**Speaker**: Alexey Tochin (UiB)

**Title**: A general approach to Schramm-Löwner Evolution and its coupling to conformal field theory.

**Abstract**: Schramm-Löwner Evolution (SLE) is a stochastic process that has made it possible to describe analytically the scaling limits of several two-dimensional lattice models in statistical physics. We consider a generalized version of SLE and then its coupling with another random object, called Gaussian free field, introduced recently by S. Sheffield and J. Dubedat. We investigate what other types of the generilized SLE can be coupled in a similar manner.

**2015**

November 24th and December 1st

**Speaker**: Anastasia Frolova, PhD student, Department of Mathematics, UiB.

**Title**: Polynomial-lemniscates, trees and braids.

**Abstract**: Following the papers "Polynomial-lemniscates, trees and braids"(Catanese, Paluszny) and "The fundamental group of generic polynomials"(Catanese, Wajnryb) we discuss the structure of the set of lemniscate-generic polynomials, i.e. polynomials whose critical level sets contain figure-eights. We describe characterization of such polynomials by trees and the action of the braid group on them.

November 10th

**Speaker**: Pavel Gumenyuk, Associate Professor, University of Stavanger.

**Title**: Loewner-type representation for conformal self-maps of the disk with prescribed boundary fixed points.

**Abstract**: It is well-known that the classical Loewner Theory provides the so-called Parametric Representation for the much studied class S of all normalized univalent holomorphic functions in the unit disk via solutions of a controllable ODE, known as the (radial) Loewner differential equation. It is less widely known that the radial Loewner equation also gives a representation of all univalent holomorphic self-maps of the unit disk with a fixed point at the origin. The corner stone of this representation is the fact that such maps form a semigroup w.r.t. the composition operation. Representations using the same heuristic scheme have been obtained for some other semigroups. The main problem in making this into a more or less general theory is that no method is known to determine whether the subsemigroup formed by all representable elements coincides with the original semigroup. Hence, it would be interesting to analyze Loewner's scheme in many different concrete examples. In this talk, we consider semigroups of univalent holomorphic self-maps with prescribed boundary regular fixed points (BRFPs). Probably the first attempt to construct a Loewner-type parametric representation for the case of one BRFP was made by H.~Unkelbach [Math. Z. 46 (1940) 329--336]. In a rigorous way it was established only in 2011 by V.V.~Goryainov [Mat. Sb.]. We discuss the case of several BRFPs, in which the approach by Goryainov cannot be applied.

November 5th

**Speaker**: Olga Vasilieva, Department of Mathematics, Universidad del Valle, Cali – COLOMBIA.

**Title**: Optimal Control Theory and Dengue Fever

**Abstract**: Dengue is a viral disease principally transmitted by Aedes aegypti mosquitoes. There is no vaccine to protect against dengue; therefore, dengue morbidity can only be reduced by appropriate vector control measures, such as: - suppression of the mosquito population, - reduction of the disease transmissibility. This presentation will be focused on implementation of these external control actions using the frameworks of mathematical modeling and control theory approach. In the first part, I will present and endemo-epidemic model derived from registered dengue case in Cali, Colombia and then propose a set of optimal strategies for dengue prevention and control. In the second part, I will present an alternative and unconventional vector control technique based on the use of biological control agent (Wolbachia) and formulate a decision-making model for Wolbachia transinfection in wild Aedes aegypti populations.

November 3rd

**Speaker**: Dmitry Khavinson, Distinguished Professor, Department of Mathematics, University of South Florida, Tampa, USA.

**Title**: Isoperimetric "sandwiches" and some free boundary boundary problems via approximation by analytic and harmonic functions

**Abstract**: The isoperimetric problem, posed by the Greeks, proposes to find among all simple closed curves the one that surrounds the largest area. The isoperimetric theorem then states that the curve is a circle. It is frst mentioned in the writings of Pappus in the third century A.D. and is attributed there to Zenodorus. However, a rigorous proof was only achieved towards the end of the 19th century! I will start by discussing some of the history of the problem and several classical proofs of the isoperimetric inequality ( e.g., those due to Steiner, Hurwitz and Carleman).. Then we shall move on to a larger variety of isoperimetric inequalities, as , e.g., in Polya and Szego classic book of 1949, but deal with them via a relatively novel approach based on approximation theory. Roughly speaking, this approach can be characterized by a recently coined term`` sandwiches". A certain quantity is introduced, usually as a degree of approximation to a given simple function, e.g., z* , |x|^2, by either analytic or harmonic functions in some norm. Then, the estimates from below and above of the approximate distance are obtained in terms of simple geometric characteristics of the set, e.g., area, perimeter, capacity, torsional rigidity, etc. The resulting ``sandwich" yields the relevant isoperimetric inequality. Many of the classical isoperimetric problems studied this way lead to natural free boundary problems for PDE, many of which remain unsolved today. Then, as an example, I will talk about some applications to the study of shapes of electrifed droplets and small air bubbles in fluid flow. During the talks I will try not only to survey the known results and methods but focus especially on many open problems that remain. This series of talks is going to be definitely accessible to the first year graduate students, or advanced undergraduates majoring in mathematics and physics who have had a semester course in complex analysisvand a routine course in advanced calculus.

October 27th

**Speaker**: Dmitry Khavinson, Distinguished Professor, Department of Mathematics, University of South Florida, Tampa, USA.

**Title**: Isoperimetric "sandwiches" and some free boundary boundary problems via approximation by analytic and harmonic functions

**Abstract**: The isoperimetric problem, posed by the Greeks, proposes to find among all simple closed curves the one that surrounds the largest area. The isoperimetric theorem then states that the curve is a circle. It is frst mentioned in the writings of Pappus in the third century A.D. and is attributed there to Zenodorus. However, a rigorous proof was only achieved towards the end of the 19th century! I will start by discussing some of the history of the problem and several classical proofs of the isoperimetric inequality ( e.g., those due to Steiner, Hurwitz and Carleman).. Then we shall move on to a larger variety of isoperimetric inequalities, as , e.g., in Polya and Szego classic book of 1949, but deal with them via a relatively novel approach based on approximation theory. Roughly speaking, this approach can be characterized by a recently coined term`` sandwiches". A certain quantity is introduced, usually as a degree of approximation to a given simple function, e.g., z* , |x|^2, by either analytic or harmonic functions in some norm. Then, the estimates from below and above of the approximate distance are obtained in terms of simple geometric characteristics of the set, e.g., area, perimeter, capacity, torsional rigidity, etc. The resulting ``sandwich" yields the relevant isoperimetric inequality. Many of the classical isoperimetric problems studied this way lead to natural free boundary problems for PDE, many of which remain unsolved today. Then, as an example, I will talk about some applications to the study of shapes of electrifed droplets and small air bubbles in fluid flow. During the talks I will try not only to survey the known results and methods but focus especially on many open problems that remain. This series of talks is going to be definitely accessible to the first year graduate students, or advanced undergraduates majoring in mathematics and physics who have had a semester course in complex analysisvand a routine course in advanced calculus.

October 20th

**Speaker**: Dmitry Khavinson, Distinguished Professor, Department of Mathematics, University of South Florida, Tampa, USA.

**Title**: Isoperimetric "sandwiches" and some free boundary boundary problems via approximation by analytic and harmonic functions

**Abstract**: The isoperimetric problem, posed by the Greeks, proposes to find among all simple closed curves the one that surrounds the largest area. The isoperimetric theorem then states that the curve is a circle. It is frst mentioned in the writings of Pappus in the third century A.D. and is attributed there to Zenodorus. However, a rigorous proof was only achieved towards the end of the 19th century! I will start by discussing some of the history of the problem and several classical proofs of the isoperimetric inequality ( e.g., those due to Steiner, Hurwitz and Carleman).. Then we shall move on to a larger variety of isoperimetric inequalities, as , e.g., in Polya and Szego classic book of 1949, but deal with them via a relatively novel approach based on approximation theory. Roughly speaking, this approach can be characterized by a recently coined term`` sandwiches". A certain quantity is introduced, usually as a degree of approximation to a given simple function, e.g., z* , |x|^2, by either analytic or harmonic functions in some norm. Then, the estimates from below and above of the approximate distance are obtained in terms of simple geometric characteristics of the set, e.g., area, perimeter, capacity, torsional rigidity, etc. The resulting ``sandwich" yields the relevant isoperimetric inequality. Many of the classical isoperimetric problems studied this way lead to natural free boundary problems for PDE, many of which remain unsolved today. Then, as an example, I will talk about some applications to the study of shapes of electrifed droplets and small air bubbles in fluid flow. During the talks I will try not only to survey the known results and methods but focus especially on many open problems that remain. This series of talks is going to be definitely accessible to the first year graduate students, or advanced undergraduates majoring in mathematics and physics who have had a semester course in complex analysisvand a routine course in advanced calculus.

October 13th

**Speaker:** Catherine Beneteau, Associate Professor, MathematicalDepartment, University of South Florida, Tampa, USA.

**Title**: Polynomial Solutions to an Optimization Problem in Classical Analytic Function Spaces

**Abstract**: In this series of talks, I will introduce some classical spaces of analytic functions in the unit disk in the complex plane called Dirichlet type spaces. Examples of these spaces include the Hardy space (functions whose coefficients are square summable), the Bergman space (functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (functions whose image has finite area, counting multiplicity). I will discuss polynomials that solve an optimization problem that I will describe. These polynomials are intimately connected to some classical tools in analysis: reproducing kernels and orthogonal polynomials. In particular, I will examine the clusters of the zeros of these optimal polynomials and show how their location depends on the space being considered. I will begin by introducing all notation and terms. The series of talks will be accessible to advanced undergraduate and beginning graduate students.

October 6th

**Speaker**: Catherine Beneteau, Associate Professor, Mathematical Department, University of South Florida, Tampa, USA.

**Title**: Polynomial Solutions to an Optimization Problem in Classical Analytic Function Spaces

**Abstract**: In this series of talks, I will introduce some classical spaces of analytic functions in the unit disk in the complex plane called Dirichlet type spaces. Examples of these spaces include the Hardy space (functions whose coefficients are square summable), the Bergman space (functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (functions whose image has finite area, counting multiplicity). I will discuss polynomials that solve an optimization problem that I will describe. These polynomials are intimately connected to some classical tools in analysis: reproducing kernels and orthogonal polynomials. In particular, I will examine the clusters of the zeros of these optimal polynomials and show how their location depends on the space being considered. I will begin by introducing all notation and terms. The series of talks will be accessible to advanced undergraduate and beginning graduate students.

September 22nd

**Speaker**: Catherine Bénéteau.

**Title:** Polynomial Solutions to an Optimization Problem in Classical Analytic Function Spaces

**Abstract:** In this series of talks, I will introduce some classical spaces of analytic functions in the unit disk in the complex plane called Dirichlet type spaces. Examples of these spaces include the Hardy space (functions whose coefficients are square summable), the Bergman space (functions whose modulus squared is integrable with respect to area measure over the whole disk), and the (classical) Dirichlet space (functions whose image has finite area, counting multiplicity). I will discuss polynomials that solve an optimization problem that I will describe. These polynomials are intimately connected to some classical tools in analysis: reproducing kernels and orthogonal polynomials. In particular, I will examine the clusters of the zeros of these optimal polynomials and show how their location depends on the space being considered.

September 15th

**Speaker:** Alex Himonas, University of Notre Dame, Notre Dame, USA.**Title**: The Cauchy problem for weakly dispersive and dispersive equations with analytic initial data.

**Abstract**: This talk presents an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces, which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution. Then, using this theorem it discusses the Cauchy problem for Camassa-Holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line. Also, it studies the continuity of the data-to-solution map in spaces of analytic functions. Finally, it compares these results with corresponding results for KdV type equations.

September 8th

**Speaker**: Kenro Furutani, Professor, Department of Mathematics, Tokyo University of Science.

**Title**: Geometry of Symmetric Operator.

**Abstract**: I will introduce two quantities, one is the Maslov index and the other is the Spectral flow. The former can be thought of as a classical mechanical quantity. The later can be seen as a quantum mechanical quantity, so that it exists only in the infinite dimension. Then I will explain their coincidence in a context of a selfadjoint elliptic boundary value problem, which is the meaning of the title.

September 1st.

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB.

**Title:** Definition of boundary complex and corresponding box operator.

**Abstract:** Last time we introduced Dolbeault complex and box operator on a complex manifold. We will use the same ideas to introduce the boundary box operator for C-R manifolds. After discussing general ideas we will make concrete calculations for the boundary of the Siegel upper half space, that is isomorphic to the Heisenberg group. We will calculate the boundary box operator in terms of left invariant vector fields of the Heisenberg group and discuss the fundamental solution for boundary box operator.

August 25th

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB.

**Title**: Solution of inhomogeneous Cauchy - Riemann equation

**Abstract**: This is an introduction lecture to the method of "orthogonal projections" used in the theory of inhomogeneous Cauchy - Riemann equations on complex manifolds. We will introduce the Dolbeault complex and calculate the box operator, which is analogue of the Laplace operator. This lecture will be used as an introduction to the boundary complex and solution of similar problems on CR manifolds.

May7th

**Speaker**: Armen Sergeev, Professor, Steklov Mathematical Institute, Moscow, Russia**Title**: Harmonic maps and Yang-Mills fields.**Abstract**: We consider a connection between harmonic maps of Riemann surfaces and Yang–Mills fields on R^4. Harmonic map from a Riemann surface into a Riemannian manifold is the extremal of the energy functional given by the Dirichlet integral. Such maps satisfy nonlinear elliptic equations of 2-nd order, generalizing Laplace–Beltrami equation. In the case when the target Riemannian manifold is Kaehler, i.e. provided with a complex structure compatible with Riemannian metric, the holomorphic and anti-holomorphic maps realize local minima of the energy. We are especially interested in harmonic maps of the Riemann sphere called briefly harmonic spheres. The Yang–Mills fields on R^4 are the extremals of the Yang–Mills action functional. Local minima of this functional are given by instantons and anti-instantons. There is an evident formal similarity between the Yang–Mills fields and harmonic maps and after Atiyah’s paper of 1984 it became clear that there is a deep reason for such a similarity. Namely, Atiyah has proved that for any compact Lie group G there is a bijective correspondence between the gauge classes of G-instantons on R^4 and based holomorphic spheres in the loop space ΩG of G. This theorem motivates the harmonic spheres conjecture stating that it should exist a bijective correspondence between the gauge classes of Yang–Mills G-fields on R^4 and based harmonic spheres in ΩG. In our talk we discuss this conjecture and possible ways of its proof.

May 6th

**Speaker**: Erlend Grong, Postdoc, Mathematical Department, University of Luxembourg.**Title**: Horizontal holonomy with applications.**Abstract**: We look at holonomy groups defined by parallel transport along curves that are tangent with respect to a given subbundle. This simple idea turns out to have powerful applications to the theory of foliations and connections on fiber bundles. It is also computable, in the sense that we can formulate a generalization of the Ambrose-Singer theorem and the Ozeki theorem for this holonomy. These results are based on joint work with Yacine Chitour (L2S, Paris XI), Frédéric Jean (ENSTA Paris Tech) and Petri Kokkonen (Varian Medical Systems, Helsinki).

May 5th

**First speaker**: Stine Marie Eik, student, Mathematical Department, UiB.

**Title**: Rademacher's Theorem

**Abstract**: The theorem of Rademacher states that a special class of continuous functions, namely Lipschitz continuous functions, is differentiable almost everywhere. In this talk we will begin by developing the necessary theory to state the theorem, for thereafter give a sketch of the proof.

**Second Speaker**: Anja Eidsheim, student, Mathematical Department, UiB.

**Title**: Hausdorff Measure and the Dimension of Pretty Pictures

**Abstract**: This talk will be starting out with reviewing the Hausdorff outer measure and a few of its properties as a function of some important parameters, particularly the Hausdorff dimension. We will proceed to look at some sets in R^n that have fractional dimension, and how the dimension of such fractals sets are calculated. If time permits, estimating the fractal dimension of naturally occurring objects studied in other fields than mathematics might also be mentioned as a group of applications of this theory.

April 28th

**Speaker**: Anastasia Frolova, PhD student, Mathematical Department, UiB.

**Title**: Quadratic differentials, graphs and Stasheff polyhedra

**Abstract**: We give a short overview of properties of rational quadratic differentials, which give solutions to certain problems in potential and approximation theory. An important problem in this context is to characterize and describe quadratic differentials with short trajectories. For this purpose we introduce a graph representation of rational quadratic differentials with one pole and classify graphs corresponding to quadratic differentials with short trajectories. We show connection between the graphs and Stasheff polyhedra, which leads to the description of the combinatorial structure of the set of quadratic differentials with short trajectories.

April 21st

**Speaker**: Mauricio Antonio Godoy Molina, Postdoc, Mathematical Department, UiB.

**Title**: Riemannian and Sub-Riemannian geodesic flows

**Abstract**: Sub-Riemannian (sR) geometry is very different from Riemannian (R) geometry in many senses, and not just an extra "sub-" in the name. Besides striking differences between their metric structures, many of the geometric invariants for R and for sR manifolds that might look quite similar in spirit, sometimes have completely unrelated behaviors. This is indeed the case for the (R and sR) geodesic flows, although in some well-studied situations (e.g., certain kinds of Lie group actions) the extra structure implies very nice relations between these flows. The goal of this talk is to show that the geodesic flows of a sR metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This helps us to describe the geodesic flow of sub-Riemannian metrics on totally geodesic Riemannian submersions. As a consequence we can characterize sub-Riemannian geodesics as the horizontal lifts of projections of Riemannian geodesics. This talked is based on a joint preprint with E. Grong available at http://arxiv.org/abs/1502.06018.

April 14th

**Speaker**: Sigmund Selberg, Professor, Mathematical Department, UiB.

**Title**: The Dirac-Klein-Gordon equations: their non-linear structure and the regularity of their solutions.

**Abstract**: The Dirac-Klein-Gordon system is a basic model of particle interactions in physics. Mathemathically this model can be studied as a system of non-linear dispersive PDEs. In this talk I will discuss the non-linear structure of these equations and how this structure enters into the analysis of the regularity properties of the solutions. If time permits I may also touch upon some themes related to Fourier restriction.

April 7th

**Speaker**: Alexey Tochin, PhD sudent, UiB.

**Title**: Rigged Hilbert spaces

**Abstract**: One of the basic result of finite-dimensional linear algebra states that for any self-adjoint or unitary operator there exist a complete system of eigenvectors. The situation becomes more complicated upon passing to the infinite-dimensional case. We consider some examples and possible solutions that involve a construction called Rigged Hilbert space. The same thing appears in attempts to define normally distributed Gaussian random law in an infinite dimensional linear space. We will try to make the talk understandable for bachelor and master level students.

March 24th

**Speaker**: Alexander Vasiliev, Professor, Mathematical Department, UiB.**Title**: Analysis and topology of polynomial lemniscates III.**Abstract**: The first part of my talk is dedicated to fingerprinting polynomial lemniscates. It turns out that they represent a good tool of image analysis and can be used for approximation of planar shapes. We study their fingerprints (in Mumford's terminology) by conformal welding and reveal the geometric sense of their inflection points. The second part concerns with the topological aspects of lemniscates. In particular, they perform the structure of an operad.

March 17th

**Speaker**: Alexander Vasiliev, Professor, Mathematical Department, UiB.**Title**: Analysis and topology of polynomial lemniscates II.**Abstract**: The first part of my talk is dedicated to fingerprinting polynomial lemniscates. It turns out that they represent a good tool of image analysis and can be used for approximation of planar shapes. We study their fingerprints (in Mumford's terminology) by conformal welding and reveal the geometric sense of their inflection points. The second part concerns with the topological aspects of lemniscates. In particular, they perform the structure of an operad.

March 10th

**Speaker**: Alexander Vasiliev, Professor, Mathematical Department, UiB.**Title**: Analysis and topology of polynomial lemniscates I.**Abstract**: The first part of my talk is dedicated to fingerprinting polynomial lemniscates. It turns out that they represent a good tool of image analysis and can be used for approximation of planar shapes. We study their fingerprints (in Mumford's terminology) by conformal welding and reveal the geometric sense of their inflection points. The second part concerns with the topological aspects of lemniscates. In particular, they perform the structure of an operad.

March 3rd

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB.**Title**: Rectifiable sets in the Euclidean space III.**Abstract**: In the last section concerning with rectifiable sets, we will review the fundamental theorem of calculus in several variables and see its generalisation on m-rectifiable sets. We also will discuss the area and co-area formulas, that can be considered as a general form of the Fubini theorem.

February 24th

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB.**Title**: Rectifiable sets in the Euclidean space II**Abstract**: Today we recall the properties of Lipschitz maps from m-dimensional Euclidean space to n-dimensional Euclidean space and prove that any m-rectifiable set in R^n is "almost" C^1-smooth manifold.

February 17th

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB.

**Title**: Rectifiable sets in the Euclidean space I

**Abstract**: In the next series of lecture (up to three) we will study the rectifiable sets in the Euclidean space. On the first lecture we recall the notion of the rectifiability of a curve and show that the one-dimensional Hausdorff measure of a Jordan curve is the length of this cirve. Then, we revise the properties of Lipschitz maps in the Euclidean space and define the rectifiable sets.

February 10th

**Speaker**: Mauricio Godoy Molina, Postdoc, Mathematical Department, UiB.**Title**: Capacity and energy III**Abstract**: To wrap up the contents from the previous seminars, I'll try to show some of the ideas behind the proof of Frostman's lemma, which relates Hausdorff measure and capacity, and the existence of sets of "arbitrary" Hausdorff dimension.

January 27th

**Speaker**: Mauricio Godoy Molina, Postdoc, Mathematical Department, UiB.

**Title**: Capacity and energy

**Abstract**: The aim of this week's seminar is to relate the $s$-capacity introduced last week with some notions in potential theory (á la Choquet), thus giving more intuitive reasons to calculate it. The fundamental idea is to characterize how big can the set of singularities of a superharmonic function be (surprise, surprise, they have capacity zero). If time permits, we will take a closer look at the proofs of some of the results mentioned last week.

January 20th

**Speaker**: Mauricio Godoy Molina, Postdoc, Mathematical Department, UiB.

**Title**: Capacity and energy

**Abstract**: The aim of this week's seminar is to introduce the Riesz $s$-capacity of a subset of Euclidean space and use it to deduce some good behavior of "small" sets, i.e., geometrically speaking, Radon measures shouldn't be too concentrated on small regions. As a consequence of this capacitarian approach (plus some technical details left behind in prior seminars), we will obtain formulas for the Hausdorff dimension of products of sets, and we will show that for any set $A$ of Radon measure $t>0$ and any $0<s<t$, there exists a subset of $A$ with Radon measure $s$.

**2014**

November 24th

**Speaker**: Christian Autenried, PhD, Mathematical Department, UiB

**Title**: Numerical methods in mathematical finance illustraded on the Monte Carlo method.**Abstract**: We introduce the idea and application of the Monte Carlo method to approximate integrals. For that purpose we construct a pseudo random number generator for uniformly distributed random variables. Furthermore, we present the method of control variates, which is among the most effective and broadly applicable technique for improving the efficiency of Monte Carlo simulation.

November 4th

**Speaker**: Mauricio Godoy Molina, Postdoc, Mathematical Department, UiB

**Title**: Lipschitz maps.

**Abstract**: To conclude this semester's series of lectures presenting some basic concepts in geometric measure theory, I will focus on Lipschitz maps. I will explain what role do Lipschitz maps play in this measure-theoretic context, how far are they from being differentiable, what is the measure of the set of critical points of a Lipschitz map and what is the relation between them and the Hausdorff measure.

October 29th and 30th

**Speaker**: Bruno Franchi, Professor, University of Bologna

**Title**: Differential forms in Carnot groups

**Abstract**: The aim of these talks is to present a comprehensive introduction to the theory of differential forms in Carnot groups (the so-called Rumin’s complex). Main topics will be:

- Left invariant differential forms in Carnot groups and the notion of weight; - The algebraic part of the differential and its pseudo-inverse; - Rumin’s classes $E_0^*$;- The complex $(E^*,d)$ of ``lifted forms’’; - Rumin differential $d_c$;- Rumin’s complex $(E_0^*,d_c)$ is homotopic to the de Rham complex;- Examples: Heisenberg groups, Engel’s group, free Carnot groups. - The intrinsic Laplacian on forms.

October 28th

**Speaker**: Dante Kalise, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria.

**Title**: Hamilton-Jacobi equations in optimal control: theory and numerics.

**Abstract**: In this talk we will review some classical and recent results concerning the link between Hamilton-Jacobi equations and optimal control, its numerical ap- proximation, and different applications. A standard tool for the solution of optimal control problem is the applicati- on of the Dynamic Programming Principle proposed by Bellman in the 50’s. In this context, the value function of the optimal control problem is characterized as the solution of a first-order, fully nonlinear Hamilton-Jacobi-Bellman (HJB) equation. The solution is understood in the viscosity solution sense introduced by Crandall and Lions. A major advantage of the approach is that a feedback mapping connecting the current state of the system and the optimal control can be obtained by means of the Pontryagin principle. However, since the HJB equation has to be solved in a state space of the same dimension as the system dynamics, the approach is only feasible for low dimensional dynamics. In the first part of the talk, we will present the main results related to HJB equati- ons, viscosity solutions and links to optimal control. The second part will be devoted to the construction of efficient and accurate numerical schemes for the approximation of HJB equations..

October 21st

**Speaker**: Alexey Tochin, PhD student, Department of Mathematics, UiB.

**Title**: The standard Gaussian distribution on infinite-dimensional linear spaces.

**Abstract**: On the way to force some infinite-dimensional analog of the standard Gaussian measure one encounters with difficulties. In particular, it is not possible in countable-dimensional Hilbert space. We consider rigged space and Gaussian Hilbert space as possible solutions. In the second hour we introduce so-called nuclear space which is a sort of a limit of a sequence of Hilbert spaces. This concept is used in the Bochner-Minlos theorem that gives a very general approach to define not only Gaussian but even more advanced distributions. This is a one of the mathematical tools for Euclidean Quantum field theory.

October 14th

**Speaker**: Anastasia Frolova, PhD student, UiB.

**Title**: Quadratic differentials, graphs and Laguerre polynomials.

**Abstract**: We give a short introduction to rational quadratic differentials. We present graph representation of a particular type of such quadratic differentials. We show how rational quadratic differentials can be applied to study of limit zero distribution of Laguerre polynomials.

October 7th

**First speaker**: Torleif Anstensrud, PhD student, Department of Engineering Cybernetics, NTNU

**Title**: Periodic solutions of nonlinear dynamic systems: Applications in legged robotics.

**Abstract**: The talk will focus on the role of periodic solutions of nonlinear differential equations in the search for walking patterns for legged robots. I will talk briefly about periodic trajectories in general, and then specifically go into detail about periodic solutions of hybrid systems (systems having both continuous and discrete dynamics) and their relationship to stable walking. Following this, the method of virtual holonomic constraints will be introduced as a tool for searching for certain periodic trajectories. The method will be demonstrated in broad terms on one of the simplest walking machines, the 2D passive biped.

**Second speaker**: Sergey Kolyubin, Research Fellow, Department of Engineering Cybernetics, NTNU

**Title**: In Pursuit of the Optimal Trajectory for Robotic Pitche

**Abstract**: We will discuss in which way the optimization can be used programming the robotic ball pitching world champion. The main challenge is planning the longest possible pitch given range, speed, and torque constraints for every joint. The original approach will be presented as a tool for trajectory generation. KUKA LWR compliant and redundant robotic arm is considered as a test bed for implementation. Finally, I will give propositions on how the results can be advanced and how you can participate there. See also the attached file for robot model.

**September 23rd**

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB**Title**: Hausdorff measure.**Abstract**: We will introduce the general construction of Caratheodory of the outer measure and shows that it is Borel regular measure on Borel sets. The particular case of that construction is the Hausdorff measure in an arbitrary metric space. We compare the Hausdorff measure with the Lebesgue measure in the Euclidean space and introduce the Hausdorff dimension of a set. As an application we calculate the dimension of the Koch snowflake.

September 16th

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB**Title**: Differentiating of measures.**Abstract**: We will defined the derivative of one measure with respect to another and will study when this derivative exists. As a corollary we obtain the generalised first theorem of calculus and Radon-Nikodim theorem. We also will discuss the Hardy-Littlwood maximal function and the Caratheodory construction of outer measures, that particularly leads to the Hausdorff measure.

September 9th

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB **Title**: Review on measure theory. **Abstract**: Last time we spoke mostly on the Lebesque measure in Euclidean space. So we need several notions such as Borel and Radon measures, regular measure, and other to extend the Vitali's covering lemma from Lebesque measure to more general measures, such as Radom measures, for example. I will give necessary definitions and examples. If we have time we start to speak about differentiation of one measure with respect to another.

September 2nd

**Speaker**: Irina Markina, Professor, Mathematical Department, UiB

**Title**: Covering lemmas in Euclidean space.

**Abstract**: This semester we organise during the analysis seminar a special course in geometric measure theory following the book by Pertti Mattila "Geometry of Sets and Measures in Euclidean spaces, fractals and rectifiability". In the 1 lecture we will consider the Vitali's and Besicovitch's covering lemmas that allows to prove some theorems about differentiability of Lebesgue and Radon measures. I will smoothly introduce the necessary material, such that not very prepared students can follow the course.

August 28th

**Speaker**: Timothy Candy, Chapman Fellow, Imperial College London, UK**Title**: Critical well-posedness for the Cubic Dirac equation**Abstract**: We outline recent work towards a global well-posedness theory for the massless cubic Dirac equation for small, scale invariant data in spatial dimensions n = 2, 3. The main difficulty is the lack of available Strichartz estimates for the Dirac equation in low dimensions. To overcome this, there are two main steps. The first is a construction of the null frame spaces of Tataru that is adapted to the Dirac equation, and which form a suitable replacement for certain missing endpoint Strichartz estimates. The second is a number of bilinear and trilinear estimates that exploit subtle cancellations in the structure of the cubic non-linearity. This is joint work with Nikolaos Bournaveas.

August 26th

**First speaker**: Kenro Furutani, Professor, Department of Mathematics, Tokyo University of Science, Tokyo, Japan**Title**: Towards a construction of the heat kernel for a higher step Grushin operator**Abstract**: I start this talk from a geometric and group theoretical introduction of Grushin type operators and a possible integral form of their heat kernel. Then I explain a construction of an action function and a candidate of a volume function associated to a higher step Grushin operator by means of complex Hamilton-Jacobi method.

**Second speaker**: Mitsuji Tamura, Assistant Professor, Department of Mathematics, Tokyo University of Science, Tokyo, Japan**Title**: On the global Carleman estimate and its applications.**Abstract**: In this talk, we consider the global Carleman estimate for inhomogeneous Schroedinger operator which depends on time. We reveal the relation between strongly and weakly pseudo convexity condition and the geometry of the domain of the definition of the Schroedinger operator. We mention applications of this inequality to the inverse problem, UCP of the boundary value problem for Schroedinger equation and to control problems.

June 10th

**Speaker**: Georgy Ivanov, PhD student, University of Bergen.

**Title**: Gaussian free field and slit stochastic flows.

**Abstract**: Connections between the Gaussian free field and SLE were first established by Schramm and Sheffield, and since then have been extensively studied in the literature.It was realized recently that the chordal, radial and dipolar SLEs are special cases of slit holomorphic stochastic flows. We investigate what other types of general slit holomorphic stochastic flows can be related to the Gaussian free field in a similar manner.

June 3rd

**Speaker**: Melkana A. Brakalova, Associate ProfessorDepartment of Mathematics, Fordham University, NewYork, USA**Title**: Conformal invariants and Teichmuller's Modulsatz in the plane**Abstract**: Module of a quadrilateral/doubly connected domain, and extremal length of a family of curves, are two interconnected conformal invariants that have played a fundamental role in the study of analytic and geometric properties of quasiconformal mappings in the plane. I will introduce these notions, discuss some of their properties and methods of evaluation as well as examples of the impact they have had.In the second part of the talk I will state and prove Teichmuller's Modulsatz, using two methods, one based on conformal mapping techniques and the other, using appropriate admissible function, which allows the possibility of extending the Modulsatz to more general settings. Applications of the Modulsatz may also be discussed.

May 27th

**First speaker**: Giovani L. Vasconcelos, Professor,Department of Mathematics, Imperial College London, andDepartment of Physics, Federal University of Pernambuco, Recife, Brazil.**Title**: Conformal geometry in multiply connected domains: a new era of conformal mapping**Abstract**: Many important problems in two-dimensional physics can be conveniently formulated as boundary-value problems for analytic functions in the complex plane. If the domain of interest is simply or doubly connected, the problem can often be solved exactly by standard conformal mapping techniques. The situation is much more complicated, however, in the case of domains with higher connectivity because conformal mappings for such domains are notoriously difficult to obtain. In this talk, I will describe a large class of conformal mappings from a bounded circular domain to multiple-slit domains which are relevant for several physical systems. The slit maps are written explicitly in terms of the primary and secondary Schottky-Klein prime functions defined by the Schottky group associated the circular domain and its subgroups. As a first application of our theory, I will compute exact solutions for the free boundary problem corresponding to the steady motion of multiple bubbles in a Hele-Shaw cell. Time-dependent solutions for multiple Hele-Shaw bubbles will also be presented. Other possible applications in fluid dynamics (e.g. vortex dynamics around multiple obstacles), growth models (e.g. Loewner evolution in multiply connected domains), and 2D string theory (e.g. multi-loop diagrams) will be briefly discussed.

**Second speaker**: Bruno Carneiro da Cunha, Professor,Federal University of Pernambuco, Recife, Brazil**Title**: Liouville Field Theory Applied to Boundary Problems**Abstract**: Riemann's mapping theorem allows one to associate a conformal map to a connected two-dimensional domain. This idea of "averaging over conformal maps" -- Liouville Field Theory -- has had many applications in critical phenomena and string theory. In this talk I will review some applications of Liouville field theory to problems, some surprisingly, related to two dimensional quantum gravity and the role of boundary conditions.

May 20th

**Speaker**: Donatella Danielli, Perdue University, West Lafayette, USA.

**Title**: Frequency functions, monotonicity formulas, and the freeboundary in the thin obstacle problem.**Abstract**: Monotonicity formulas play a pervasive role in the study ofvariational inequalities and free boundary problems. In this talk wewill describe a new approach to a classical problem, namely the thinobstacle (orSignorini) problem, based on monotonicity properties for a family ofso-called frequency functions.

May 14th

**Speaker**: István Prause, professor of the University of Helsinki.**Title**: Bilipschitz maps, logarithmic spirals and complex interpolation.**Abstract**: How much a bilipschitz map can spiral? We explore two complementary aspects: how fast and how often. Quasiconformal techniques turn out to be effective to study this problem. In many ways, rotational phenomena for bilipschitz maps are dual to stretching properties of quasiconformal maps. I will contrast these two and explain what links them together.The talk is based on joint work with K. Astala, T. Iwaniec and E. Saksman.

May 6th

**Speaker**: Mark Agranovsky, professor, Bar Ilan University (Israel).

**Title**: Common nodal surfaces in Euclidean space.

**Abstract**: Nodal sets are loci of Laplace eigenfunctions. Theyfairly desicribe wave propagation and are subject of a stronginterest. While the global construction of a singe nodal sethardly can be well understood, one may hope that common nodalsets of large families of eigenfunctions must have a prettyspecial geometry. In particular, it was conjectured thatcommon nodal hypersurfaces of eigenfunctions,arising as thespectrum of a compactly supported function, arecones-translates of the zero sets of nonzero harmonichomogeneous polynomials (spatial harmonics). It is confirmedin 2d and is still open in higher dimensions. The approachesand the current status of the problem will be discussed.

April 29th

**Speaker**: Christian Autenried, PhD student, UiB**Title**: Classification of 2-step nilpotent Lie algebras**Abstract**: We will describe some metric approach to study Lie algebras that are nilpotent of step 2. I will try to make my talk understandable for bachelor and master level students.

April 8th

**Speaker**: Alexey Tochin, PhD student, UiB

**Title**: Gaussian free field and Schramm-Loewner evolution

**Abstract**: This is a continuation of the talk about Gaussian free field presented in Geilo. We begin with reminding the definition and the basic properties. Then we see how the zero level line of the approximation of Gaussian free field generates the same random law as the one from the Scramm-Loewner evolution (O. Schramm and S. Sheffield 2005). We continue with the Markov property of the Gaussian free field to illustrate that fact. In the end the so-called Ward identities will be discussed.

April 1st

**Speaker**: Ragnar Winther, professor, University of Oslo.

**Title**: Local bounded cochain projections and the bubble transform.

**Abstract**: The study of discretizations of Hodge Laplace problems in finite element exterior calculus unifies the theory of mixed finite element approximations of a number of problems in areas like electromagnetism and fluid flow. The key tool for the stability analysis of these discretizations is the construction of projection operators which commute with the exterior derivative and at the same time are bounded in the proper Sobolev norms. Such projections are referred to as bounded cochain projections. The canonical projections, constructed directly from the degrees of freedom, will commute with the exterior derivative, but unfortunately, they are not properly bounded. On the other hand, bounded cochain projections have been constructed by combining a smoothing operator and the unbounded canonical projection. However, an undesired property of these smoothed projections is that, in contrast to the canonical projections, they are nonlocal. Therefore, we have recently proposed an alternative construction of bounded cochain projections, which also is local. This construction can be seen as a variant of the well known Clément operator, and it utilizes a double complex structure defined on the macroelements associated the subsimplexes of the grid. In addition, we will also discuss a new tool for analysis of finite element element methods, referred to as the bubble transform. In contrast to all the projections operators above, this transform will lead to projections with bounds which are independent of the polynomial degree of the finite element spaces. As a consequence, this can potentially simplify the analysis of the so-called p-method.

March 18th

**Speaker**: Mauricio Godoy Molina, postdoc, UiB

**Title**: Abstracting the rolling problem.

**Abstract**: The aim of this seminar is to present a generalization of the rolling system to the abstract framework of Cartan geometries, which are the most general environment in which the notion of "development" can be carried out. In this new context, many of the seemingly ad-hoc geometric concepts introduced for the rolling system become somewhat more natural, albeit less intuitive.This talk is based on joint work with Y. Chitour and P. Kokkonen from Paris XI.

March 11th

**Speaker**: Alexander Vasiliev, professor, University of Bergen.

**Title**: Loewner equation and integrable systems.

**Abstract**: Abstract: We argue that the Loewner equation serves asa background tool for some integrable systems. In particular,splitting time leads to the Vlasov equation for the distributionfunction of plasma, and to the Benney hierarchy. We also show thatthe solution to the Loewner equation with infinite dimensional time givesthe Lax function which solves the dispersionless KP hierarcy inwhich the Benney equations may be recovered as the second inthe dKP hierarchy.Joint work with Dmitri Prokhorov and Maxim Pavlov.

February 4th

**Speaker**: Victor Gichev, Sobolev Institute of Mathematics (Omsk Branch), Omsk, Russia.

**Title**: Invariant cone fields and semigroups in Lie groups.

**Abstract**: I shall describe briefly some areas of mathematics related to the objects of the title and concentrate on one of them, bi-invariant orderings in Lie groups. I'll formulate a theorem which characterizes cones corresponding to the ''good'' orderings and begin preparations to its proof. This includes a theorem on reachable sets of an invariant control system in a nilpotent Lie group extended by R.

January 28th.

**Speaker**: Nam-Gyu Kang, Seoul National University, Republic of Korea.

**Title**: Gaussian Free Field, Conformal Field Theory, and Schramm–Loewner Evolution.

**Abstract**: I will present an elementary introduction to conformal field theory in the context of complex analysis and probability theory. Introducing Ward functional as an insertion operator under which the correlation functions are transformed into their Lie derivatives, I will explain several formulas in conformal field theory including Ward's equations. This presentation will also include relations between conformal field theory and Schramm–Loewner evolutions in various conformal types. Some recent work in the case of multiple SLE curves and their classical limits will be discussed. This is joint work with Nikolai Makarov, Hee-Joon Tak, Dapeng Zhan, and Tom Alberts.

January 21st

**Speaker**: Dmitri Prokhorov, professor, Saratov State University, Russia.

**Title**: On a ratio of harmonic measures of slit sides.

**Abstract**:The talk is devoted to estimates of a ratio for harmonic measures of slit sides depending ongeometric properties of a slit. For a domain $\Omega$ slit along a curve $\gamma=\gamma[0,t]$ andfor a point $a\in\Omega\setminus\gamma$, define $m_k(t)$, $k=1,2$, harmonic measures of$\gamma_k[0,t]$ at $a$ with respect to $\Omega$, where $\gamma_1[0,t],\gamma_2[0,t]$ are two sidesof $\gamma[0,t]$. We estimate asymptotically $$\frac{m_1(t)}{m_2(t)},\;\;\;t\to+0,$$ taking intoaccount a geometry of $\gamma$.

**2013**

November 19th

**Speaker**: Alexey Tochin, PhD student, University of Bergen.

**Title**: General 1-Slit Loewner Equation.

**Abstract**: We introduce a family of equations which in a certain sense generalize various versions of the Loewner equation. We start with the definitions of very general objects on an arbitrary Riemann surface. Using them, we introduce and analyze the 1-Slit Loewner Equation. We restrict our attention to the case of the hyperbolic Riemann surface (the unit disk) and explain some of our new results.This is part of our ongoing joint work with G.Ivanov and A.Vasiliev.

November 14th

**Speaker**: Hans Martin Reimann, University of Bern, Switzerland

**Title**: The mathematics of hearing

**Abstract**: Many mathematical problems arise in the study of the auditory pathway. The focus will be on two basic topics:How is the signal processing done in the inner ear and in the peripheral acoustic centers?Is there a calculus that suitably describes the neuronal processes?

November 12th

**Speaker**: Georgy Ivanov, PhD student, Mathematical Department, UiB

**Title**: Stochastic holomorphic semiflows in the unit disk.

**Abstract**: In 1984, H.Kunita considered stochastic flows on smooth paracompact manifolds. In particular, he showed that if the vector fields X_1, ... X_m defining a Stratonovich SDE are complete and generate a finite-dimensional Lie algebra G, then the corresponding flow is in fact a flow of diffeomorphisms, taking values in G.We restrict ourselves to the case of holomorphic flows on the unit disk, but allow the vector field at "dt" to be semicomplete. In this case, General Loewner theory provides an immediate proof of the fact that the corresponding stochastic flow is a flow of holomorphic maps of the unit disk into itself (a holmorphic semiflow). The results can be extended to multiply connected domains, as well as to general complex hyperbolic manifolds.This is part of our ongoing joint work with A.Tochin and A.Vasiliev.

November 5th

**Speaker**: Victor Kiselev, PhD student, Mathematical Department, UiB.

**Title**: A fast segmentation method for color images.

**Abstract**: Image segmentation has always been one of the central questions in image processing. To segment an image means to divide it intonon-overlapping "meaningful" regions. We will discuss some methods used for such problems and a particular semi-supervised segmentation method, which is based on feature extraction from different color spaces and employs variational framework.

October 29th

**Speaker**: Christian Autenried (PhD student, UiB).

**Title**: Classification of H-type Lie algebras

**Abstract**: In the talk we show that the extension of an H-type Lie algebra n_{r,s} induced by a Clifford algebra Cl_{r,s} by n_{8,0}, n_{4,4} or n_{0,8} is preserving isomorphisms. This implies a method which reduces the classification of an arbitrary H-type Lie algebra n_{r,s} to the classification of n_{t,u} with 0 \leq t,u \leq 8. Furthermore, we give an overview of the current state of research of the classification of n_{t,u} with 0 \leq t,u \leq 8 and some methods to classify the H-type Lie algebras n_{t,u}.

October 22nd

**Speaker**: Irina Markina, professor, University of Bergen.

**Title**: Integer lattices on pseudo-$H$ groups

**Abstract**: During the spring semester I presented our last result with A.Korolko and M.Godoy about the definition of pseudo $H$-type groups (or general $H$-type groups). At the present seminar I will explain why these groups admit an integer lattice. The existence of a lattice on a Lie group is equivalent to the existence of a basis on the corresponding Lie algebra. I will give all necessary definitions and present the construction of the concrete basis in one of pseudo $H$-type Lie algebras. This is a joint work with Professor Kenro Furutani from the University of Science and Technology of Tokyo, see arXiv:1305.6814.

October 8th

**Speaker**: Mahdi Khajeh Salehani, postdoc, University of Bergen.

**Title**: A geometric approach to nonholonomic dynamics.

**Abstract**: The Euler-Lagrange equations, while universal, are not always effective to analyze the dynamics of mechanical systems. For example, it is difficult to study the motion of a simple mechanical system like the Euler top using the Euler-Lagrange equations, either intrinsically or in generalized coordinates. In fact, Euler (1752) discovered that the equations of motion for the rigid body become significantly simpler if one uses, instead of the generalized velocities, the angular velocity components relative to a body frame.There actually exist a number of variational principles one may use to derive the equations of constrained mechanical systems. In this talk, we study some of these principles and give a geometric interpretation of the derived equations of motion in both holonomic and nonholonomic settings, generalizing the ideas pioneered by Euler and further developed by Lagrange (1788) and Poincaré (1901).

October 1st

**Speaker**: Nikolay Kuznetsov (Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems in Mechanical Engineering, Russian Academy of Sciences, St Petersburg).

**Title**: Steady water waves with vorticity: spatial Hamiltonian structure.

**Abstract**: Spatial dynamical systems are obtained for two-dimensional steady gravity waves with vorticity on water of finite depth. These systems have Hamiltonian structure and Hamiltonian is essentially the flow-force invariant.

September 24th

**Speaker**: Alexander Vasiliev, professor, University of Bergen

**Title**: Extremal metrics in the modulus problem for some families of curves and surfaces.

**Abstract**: The modulus of families of curves (or extremal length) is a powerful method in analysis introduced originally by Grötzsch (1928) and developed by Beurling and Ahlfors (1950). It enjoys conformal invariance and uniqueness of the extremal metric. However the existence of the latter is a difficult problem and its explicit expression is known only in few cases. In 1974, Rodin proposed a method of calculation of the extremal metric in the case when the family of curves is the image of another family in the plane for which the extremal metric is known. We extend this theorem for the Euclidean space and polarizable groups and propose some application to integral inequalities. This is a joint work with Irina Markina (Bergen) and Melkana Brakalova (New York).

September 17th

**Speaker**: Anastasia Frolova, PhD student, University of Bergen

**Title**: Cowen-Pommerenke type inequalities for univalent functions.

**Abstract**: We present a new estimate for angular derivatives of univalent maps at the fixed points. The method we use is based on the properties of the reduced modulus of a digon and the problem of extremal partition of a Riemann surface.

September 10th

**Speaker**: Anastasia Frolova, PhD student, University of Bergen

**Title**: One-parameter semigroups in the unit disk and estimates for angular derivatives.

**Abstract**: We introduce a technique which allows us to deduce estimates for general holomorphic functions from estimates for univalent once. The method relies on theory of semigroups of holomorphic self-mappings of the unit disc.

September 3rd

**Speaker**: Alexander Vasiliev, University of Bergen

**Title**: Boundary distortion under conformal map

**Abstract**: We survey some results on boundary distortion under conformal self-maps of the unit disk. In particular, we review Cowen-Pommerenke and Anderson-Vasiliev type inequalities making use of the moduli method.

August 27th

**Speaker**: Irina Markina, professor, University of Bergen

**Title**: Relation between the module of family of curves and family of surfaces on Carnot groups.

**Abstract**: I will start from reviewing the relation between the module of a family of curves connecting two compacts and family of surfaces separating these two compacts. I will present the extremal metrics and extremal families of curves and surfaces. Then I introduce the analogous notions of module on the Carnot groups pointing the novelties and difficulties. The main aim is to understand the extremal families and metrics in the geometrical setting of Carnot groups.

May 08th **Speaker**: Galina Filipuk, Professor, University of Warsaw**Title**: Multiple orthogonal polynomials and their properties.**Abstract**: In this talk I shall speak about the multiple orthogonal polynomials, their definition, the raising and lowering operators and the differential equations they satisfy. I shall also present a few examples: the multiple orthogonal polynomials with exponential cubic weight, their zeros and the properties of Wronskians of multiple orthogonal polynomials. This is a joint work with W. Van Assche and L. Zhang (KULueven, Belgium).

May 2nd

**Speaker**: Anastasia Frolova, PhD student, University of Bergen**Title**: Cowen-Pommerenke type inequalities for univalent functions.**Abstract**: We consider conformal maps from the unit disk into itself, such that they have two fixed points on the unit circle, and which are conformal at these points. We obtain an estimate of the product of the angular derivatives of such maps at the fixed points. The method we use is based on the properties of the reduced modulus of a digon and the problem of extremal partition of a Riemann surface. (Joint work with Alexander Vasil'ev)

April 25th

**Speaker**: Georgy Ivanov, PhD student, University of Bergen**Title**: Random walk and PdE on graphs**Abstract**: The deep connections between Brownian motion and partial differential equations are well-known. In this lecture we consider the discrete counterparts of these concepts - random walk and partial difference equations on graphs. This allows to illustrate the main ideas of the continuous theory, but at the same time requires having only elementary mathematical background (a first course in probability and basic notions of measure theory and discrete mathematics should suffice).

April 18th

**Speaker:** Bruno Franchi (Dipartimento di Matematica Universita' di Bologna, Bologna, Italy)

**Title**: Intrinsic graphs in Carnot groups

**Abstract**: he aim of this talk is to provide an introduction to the theory of intrinsic graphs in Carnot groups, and, in particular, to that of intrinsic Lipschitz graphs. The simple idea of intrinsic graph is the following one: let M, H be complementary homogeneous subgroups of a group G, then the intrinsic (left) graph of f: A\subset M\to H is the set graph f ={g . f(g): g\in A }. This notion deserves the adjective ``intrinsic'' since it is invariant under left translations or homogeneous automorphisms of the group (dilations in particular). We stress that neither Euclidean graphs are necessarily intrinsic graphs nor the opposite. Intrinsic graphs appeared naturally while studying non critical level sets of differentiable functions from a Carnot group G to the Euclidean space R^k. Indeed, implicit function theorems for groups can be rephrased stating precisely that these level sets are always, locally, intrinsic graphs. We shall discuss also a remarkably deep relationship between intrinsic graphs associated with a group decomposition and the so-called Rumin's complex (E_0^*,d_c) of differential forms in a Carnot group G.

April 11th

**Speaker**: Alexey Tochin (PhD student, UiB)

**Title**: A generalization of SLE

**Abstract**: This seminar is a continuation of the two previous ones. After repeating the key points we will proceed to the main subject, namely generalized SLE with one slit which was introduced in the very end of the last seminar. In essence, this is a two-parametric family of equations that contains the well-known Radial, Dipolar and Chordal SLE as 3 special cases. The properties of all of these new SLE equations are very similar to those of the classical ones. We will not prove them (the work is still in progress), but there will be a lot of numerical simulations illustrating them. (Joint work with Georgy Ivanov and Alexander Vasil'ev)

March 21st

**Speaker:** Georgy Ivanov (PhD student, UiB)

**Title**: A generalization of SLE

**Abstract**: Radial, Chordal and Dipolar SLE (Schramm-Loewner evolution) can be defined as families of conformally invariant measures on curves, possessing the domain Markov property. The domain Markov property is closely related to the fact that the governing equations can be represented as time-homogeneous diffusion equations. We use general Loewner theory (Bracci, Contreras, Diaz-Madrigal, Gumenyuk) and consider a more general class of diffusions which generate slit evolutions. We use SLE with attractive boundary point (constructed in our earlier paper) as a model example for this class of measures. (Joint work with Alexey Tochin and Alexander Vasil'ev).

March 14th

**Speaker:** Georgy Ivanov (PhD student, UiB)

**Title**: A generalization of SLE

**Abstract**: Radial, Chordal and Dipolar SLE (Schramm-Loewner evolution) can be defined as families of conformally invariant measures on curves, possessing the domain Markov property. The domain Markov property is closely related to the fact that the governing equations can be represented as time-homogeneous diffusion equations. We use general Loewner theory (Bracci, Contreras, Diaz-Madrigal, Gumenyuk) and consider a more general class of diffusions which generate slit evolutions. We use SLE with attractive boundary point (constructed in our earlier paper) as a model example for this class of measures. (Joint work with Alexey Tochin and Alexander Vasil'ev).

February 28th

**Speaker**:** **Mahdi Khajeh Salehani (Postdoc, UiB)

**Title**: Classical nonholonomic vs. vakonomic mechanics: a report on the 'debate'**Abstract**: To study constrained mechanical systems, there are at least two approaches one may take, namely the "classical nonholonomic approach", which is based on the Lagrange-d'Alembert principle and is not variational in nature, and a variational axiomatic one known as the "vakonomic approach".In fact there are some fascinating differences between these two procedures, e.g., they do not always give the same equations of motion; the distinction between these two procedures has a long and distinguished history going back to Korteweg (1899), and has been discussed in a more modern context by Arnold, Kozlov and Neishtadt since 1983.In this seminar, we present the classical nonholonomic mechanics and the vakonomic mechanics of systems with constraints, and will compare them in order to see when these two mechanics are equivalent, i.e., when they give the same system of equations. For the class of mechanical systems that they are not so, we determine which one of these approaches is the appropriate one for deriving the equations of (mechanically possible) motions.

February 21st

**Speaker**: Christian Autenried (PhD student, UiB)

**Title**: Clifford modules and admissible metrics**Abstract**: In this seminar we remind the definition of Clifford modules and present their classifications according to the metric that makes representations skew symmetric. Furthermore, we introduce pseudo-H-type algebras and consider three particular examples.

February 14th

**Speaker**: Arne Stray (professor, UiB)

**Title**: Approximation by polynomials and translates of the Riemann zeta function

**Abstract**: We discuss some recent work involving Mergelyans theorem and certain properties of Riemann famous function.

February 7th

**Speaker**: Alexander Vasiliev

**Title**: Introduction to Neurogeometry

**Abstract**: It will be an informal comprehensive introduction to a rather recent area of Neurogeometry. In particular, we address the problems of inpainting by anisotropic diffusion and sub-Riemannian underlying geometry of the first visual cortex.

January 31st

**Speaker**: Irina Markina, Professor, UiB

**Title**: Algebras of Heisenberg type and possible generalisations

**Abstract**: In the first seminar we introduced so called general H-type Lie algebras. On the second seminar I will reveal the relation between these Lie algebras and composition of quadratic forms and Clifford modules.

January 24th

**Speaker**: Irina Markina, Professor, UiB**Title**: Algebras of Heisenberg type and possible generalisations**Abstract**: We introduce a special class of nilpotent Lie groups of step 2, that generalises the so-called Heisenberg-type groups, defined by A. Kaplan in 1980. We change the presence of the inner product to an arbitrary scalar product and relate the construction to the composition of quadratic forms and Clifford modules. We present geodesic equations for the sub-semi-Riemannian metric on nilpotent Lie groups of step 2 and solve them for the case of general H-type groups. We discuss possible classifications of these groups.

**2012**

November 27th

**Speaker**: Prof. Boris Kruglikov (University of Tromsø, Norway)

**Title**: A tale of two G2

**Abstract**: Exceptional Lie group G2 is a beautiful 14-dimensional continuous group, having relations with such diverse notions as triality, 7-dimensional cross product and exceptional holonomy. It was found abstractly by Killing in 1887 (complex case) and then realized as a symmetry group by Engel and Cartan in 1894 (real split case). Later in 1910 Cartan returned to the topic and realized split G2 as the maximal finite-dimensional symmetry algebra of a rank 2 (non-holonomic) distribution in dimension 5. This follows from Cartan's analysis of the symmetry groups of Monge equations of the form y'=f(x,y,z,z',z"). I will discuss the higher-dimensional generalization of this fact, based on the joint work with Ian Anderson. Compact real form of G2 was realized by Cartan as the automorphism group of octonions in 1914. In the talk I will also explain how to realize this G2 as the maximal symmetry group of a geometric object (non-degenerate almost complex structure in dimension 6) and discuss what other symmetry groups are allowed.

**Speaker**: Prof. Sergey Favorov (Kharkov National University, Ukraine)

**Title**: Blaschke-type conditions on unbounded domains, generalized convexity,and applications in perturbation theory

**Abstract**: We introduce a notion of $r$-convexity for subsets of the complex plane. It is a pure geometric characteristic that generalizes the usual notion of convexity. For example, each compact subset of any Jordan curve is $r$-convex.Further, we investigate subharmonic functions that grow near the boundary in unbounded domains with $r$-convex compact complement. We obtain the Blaschke-type bounds for its Riesz measure and, in particular, for zeros of unbounded analytic functions in unbounded domains. These results are based on a certain estimatesfor Green functions on complements of some neighborhoods of $r$-convex compact set.We apply our results in perturbation theory of linear operators in a Hilbert space. Namely, let $A$ be a bounded linear operator with an $r$-convex spectrum such that the complement of its essential spectrum $\sigma_{ess}(A)$ is connected, and a linear operator $B$ be in the Schatten - von Neumann class $S_q$. We find quantitative estimates for the rate of condensation of the discrete spectrum $\sigma_d(A+B)$ near the essential spectrum $\sigma_{ess}(A)$ (note that under our condition $\sigma_{ess}(A+B)=\sigma_{ess}(A)$).

November 20th

**Speaker**: Anastasia Frolova (PhD student, UiB)

**Title**: Extremal length method and estimates of angular derivatives of conformal mappings.

**Abstract**: We introduce the notion of reduced modulus of a digon and use it to solve the following extremal problem of conformal mappings. We consider conformal maps from the unit disk into itself, such that it has a fixed point on the unit circle and is conformal at it. We obtain an estimate of the angular derivative of such maps.

November 13th

**Speaker**: Alexey Tochin (PhD student, UiB)

**Title**: On the connection between Gaussian Free Field (GFF), Stochastic Loewner Evolution (SLE) and Conformal Field Theory (CFT).**Abstract**: The main purpose of the talk is to show connections between stochastic processes, measures on curves, random 2-dimensional distributions, operator valued distributions and representations of the Virasoro algebra.We begin with a simple problem of the Harmonic Explorer and show its natural connection to SLE(4), interface of GFF and a representation of the Virasoro algebra. We will give necessary definitions via discrete versions of these objects as well as explore the continuous approach. Then turning to the general case of the Choradal SLE we will give a brief introduction to the Conformal Field Theory. We finish with the approach by Makarov and Kang designed to merge together all of the three concepts given in the title.

November 6th

**Speaker**: Christian Autenried (PhD student, UiB)

**Title**: Sub-Riemannian geometry of Stiefel manifolds.

**Abstract**: In the talk we consider the Stiefel manifold V(n;k) as a principal U(k)-bundle over the Grassmann manifold and study the cut locus from the unit element. We will give the complete description of this cut locus on V(n;1) and present the suffi{#0E}cient condition on the general case.

October 30th

**Speaker**: Alexander Vasiliev

**Title**: SLE and CFT

**Abstract**: We review connections between the Stochastic Loewner Evolution, Gaussian Free Field and Conformal Field Theory following a recent preprint by Makarov and Kanghttp://arxiv.org/abs/1101.1024

October 23rd

**Speaker**: Erlend Grong (associate professor, HiB)

**Title**: Submersions, lifted Hamiltonian systems and rolling manifolds

**Abstract**: A submersion is a map between two manifolds $\pi:Q \to M$ that is surjective on each tangent space. The kernel of this map gives us a sub-bundle called the vertical bundle. A chosen complement $H$ to this bundle is called an Ehresmann connection. If the submersion $\pi$ is between two Riemannian manifolds and it in addition has the property of being a fiber wise isometry when restricted to $H$, then geodesics in $M$ are just projections of geodesics in $Q$ which are horizontal to $H$. Conversely, if the same conditions hold for $\pi$ and $Q$ is a principal $G$-bundle over $M$ with a "sufficiently nice metric", then the projections of the geodesics in $Q$ are the trajectories of gauge-charged particles under the influence of a magnetic field. This magnetic field is represented by the curvature of $H$.We will generalize these ideas by looking at Hamiltonian systems on $M$ and we construct a lifting of them to $Q$. Then we look at what we can know of the solutions in $M$ by looking at the solutions in $Q$, and conversely, how we can describe the solutions in $Q$ by their projections to $M$. This description relies on a new idea of parallel transport of vertical vector fields.In order to show the application of this new method, we apply it to try to describe optimal curves of the rolling manifold problem.

October 9th

**Speaker:** Arne Stray (professor, UiB)

**Title:** Dominating sets and simultaneous approximation in the unit disc

**Abstract:** For a space H of analytic functions in the unit disc D, we look for a geometric characterization of the subsets F such that sup of |f| over F is equal to sup of |f| over D for all f in H. This problem is related to problems in simultaneous approximation that also will be discussed.

September 25th

**Speaker:** Georgi Raikov (Pontificia Universidad Católica de Chile)

**Title:** Resonances and spectral shift function singularities for a magnetic Schroedinger operator

**Abstract:** Let H_{0} be the 3D Schroedinger operator with constant magnetic field, V be an electric potential which decays sufficiently fast at infinity, and H = H_{0} + V. First, we consider the asymptotic behaviour of the Krein spectral shift function (SSF) for the operator pair (H,H_{0}), near the Landau levels which play the role of thresholds in the spectrum of H_{0}. We show that the SSF has singularities near the Landau levels, and describe these singularities in terms of appropriate Berezin - Toeplitz operators. Further, we define the resonances for the operator H and investigate their asymptotic distribution near the Landau levels. We show that under suitable assumptions on the potential V there are infinitely many resonances near every fixed Landau level. We find the main asymptotic term of the corresponding resonance counting function, which again is expressed in terms of the Berezin - Toeplitz operators arising in the description of the SSF singularities.

The talk is based on joint works with J.-F. Bony (Bordeaux), V. Bruneau (Bordeaux), and C. Fernández (Santiago de Chile).

Partially supported by the Chilean Science Foundation Fondecyt under Grant 1090467.

September 18th

**Speaker**: Oles Kutovyi (University of Bielefeld, Germany and MIT, USA)

**Title**: Stochastic evolutions in ecological models and their scalings**Abstract**: We analyze an interacting particle system with a Markov evolution of birth-and-death type in continuum. The corresponding Vlasov-type scaling, which is based on a proper scaling of corresponding Markov generators and has an algorithmic realization in terms of related hierarchical chains of correlation functions equations is studied. The existence of rescaled and limiting evolutions of correlation functions as well as convergence to the limiting evolution are shown.

September 11th

**Speaker**: Olga Vasilieva (Universidad del Valle, Cali, Colombia)

**Title**: Catch-to-stock dependence: the case of small pelagic fish with bounded fishing effort

**Abstract**: Small pelagic fish (such as herring, anchovies, capelin, smelts, sardines or pilchards) is characterized by high reproduction rate and rather short life-cycle. Additionally, pelagic fish stock have strong recurrent cycles of fish abundance and scarcity and may provide high catch yields per unit of fishing effort even within the scarcity periods. The latter may provoke a collapse of fish stock since our abilities to predict their periods of abundance and/or scarceness are very limited.

Empirical evidence and biological characteristics of pelagic fish suggest that, in contradiction with traditional fishery models, marginal catch of pelagic species does not react in linear way to changes in stock level. In this presentation, we allow non-linearity in catch-to-stock parameter and propose another variant of single-stock harvesting economic model focusing on the dependence of stationary solutions upon such non-linear parameter.

Our principal interest consists in finding an optimal fishing effort leading to stationary solutions that prevent fishing collapse and help to avoid the species extinction. To do so, we first formulate a social planner's problem in terms of optimal control for infinite horizon, then analyze its formal solution by applying the Pontryagin's maximum principle and finally revise a possibly of a singular arc appearance. In conclusion, we also examine some core properties of stationary equilibrium reachable by means of a singular optimal control and prove the existence and uniqueness of steady states under some additional assumptions.

This is a joint with Erica Cruz-Rivera (Universidad del Valle, Colombia) and Hector Ramirez-Cabrera (CMM, Universidad de Chile, Chile) within the frameworks of the Research Project C.I. 7807, 2010-2012.}

September 4th

**Speaker:** Georgy Ivanov (PhD student, UiB)

**Title:** Self-intersections, corners and cusps of Loewner slits.

**Abstract:** In 2010 Lind, Marshall and Rohde gave a characterization of Loewner driving terms generating Loewner traces with self-intersections and infinite spirals. Using a similar technique we characterize driving terms generating slits with corners, and propose a way to characterize driving terms generating tangent slits and slits with cusps.

August 28th

Speaker: **Mahdi Khajeh Salehani** (Postdoc, UiB)

Title:** "Controllability on infinite-dimensional manifolds"**

Abstract: One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classic result in control theory of finite-dimensional systems is Rashevsky-Chow's theorem that gives a sufficient condition for controllability on any connected manifold of finite dimension.

This result was proved independently and almost simultaneously by Rashevsky (1938) and Chow (1939).In this seminar, following the unified approach of A. Kriegl and P.W. Michor (1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Rashevsky-Chow's theorem for control systems in regular connected manifolds modeled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable.This is a joint work with Prof. Irina Markina.

May 29th

Speaker: **Vladimir Maz'ya** (Professor, University of Liverpool (UK) and University of Linköping (Sweden))

Title: "**Higher Order Elliptic Problems in Non-Smooth Domains"**

Abstract: We discuss sharp continuity and regularity results for solutions of the polyharmonic equation in an arbitrary open set. The absence of information about geometry of the domain puts the question of regularity properties beyond the scope of applicability of the methods devised previously, which typically rely on specific geometric assumptions. Positive results have been available only when the domain is sufficiently smooth, Lipschitz or diffeomorphic to a polyhedron.The techniques developed recently allow to establish the boundedness of derivatives of solutions to the Dirichlet problem for the polyharmonic equation under no restrictions on the underlying domain and to show that the order of the derivatives is maximal. An appropriate notion of polyharmonic capacity is introduced which allows one to describe the precise correlation between the smoothness of solutions and the geometry of the domain.We also study the 3D Lam\'e system and establish its weighted positive definiteness for a certain range of elastic constants. By modifying the general theory developed by Maz'ya (Duke, 2002), we then show, under the assumption of weighted positive definiteness, that the divergence of the classical Wiener integral for a boundary point guarantees the continuity of solutions to the Lamé system at this point.

May 15th

Speaker:** Simon G. Gindikin **(Rutgers University, USA)

Title:** "****Holomorphic language for Cauchy-Riemann cohomology"**

Abstract: In multidimensional complex analysis it is not possible to work just with holomorphic functions: we need also to consider higher Cauchy-Riemann cohomology. Usually for their consideration we need go outside of holomorphic analysis. It turns out that there is a purely holomorphic language for cohomology . We will talk about this language discuss several situations at Fourier analysis, representations, differential equations where it is natural to work with cohomology.

May 8th

Speaker: **Anastasia Frolova** (MSc student, UiB)

Title:** "****Critical measures and quadratic differentials"**

Abstract: In this talk, we will show how the theory of quadratic differentials is applicable to the problem of describing critical measures, which provide critical points of weighted logarithmic energy on the complex plane. We will also overview the connection between critical measures and solutions to the Lamé equation. The talk is based on the master thesis of A. Frolova

April 24th

Speaker: **Xue-Cheng Tai** (Professor, UiB)

Title: **"Partitioning of domains as a mathematical problem: numerical algorithms and applications"**

Abstract: This talk is devoted to the optimization problem of continuous multi-partitioning, or multi-labeling, which is based on a convex relaxation of the continuous Potts model. In contrast to previous efforts, which are trying to tackle the optimal labeling problem in a direct manner. Some algorithms will be supplied to numerical solve these problems with speed efficiency.

In the end, we will also present several recent algorithms for computing global minimizers based on graph cut algorithms and augmented Lagrangian approaches.

April 17th

Speaker: **Wolfram Bauer** (Georg-August-Universität Göttingen, Germany)

Title: **"Commutative Toeplitz algebras on weighted Bergman spaces over the unit ball"**.

Abstract: We recall the notion of Toeplitz operators acting on the Hardy space over the unit circle S^{1} and on weighted Bergman spaces over a domain Ω ⊂ C^{n}, respectively. Then we discuss the analysis of corresponding C^{∗}- and Banach algebras which are generated by Toeplitz operators (we call them Toeplitz algebras). In the case where Ω = D ⊂ C is the open unit disc we describe classes of commutative C^{∗}-algebras that are induces by automorphisms of D. The results can be generalized to the higher dimensional setting of standard weighted Bergman spaces over the unit ball in C^{n}, where n > 1. However in this case, new types of commutative Toeplitz Banach algebras appear that are not ∗-invariant and have no counterpart in the one-dimensional situation. If there is time we will explicitly describe the structure of the simplest type of such an algebra which arises in dimension n = 2. Some of the results have been obtained recently in a joint work with N. Vasilevski.

April 10th

Speaker:** Christian Autenried** (PhD student, UiB)

Titile: **"Chow's theorem"**

Abstract: We will discuss Wei-Liang Chow's paper "Über Systeme von linearen partiellen Differentialgleichungen erster Ordnung."(1939). This paper includes Chow's version of the Rashevski-Chow theorem. Our aim is to introduce the approach of Chow and to prove his theorem. Furthermore, you will get a translation of the paper which was only available in German.

April 3rd

Speaker: **Yacine Chitour** (Laboratoire des signaux et systèmes, Université Paris-Sud 11)

Title:** "****Rolling on a space form"**

Abstract: In this talk, we present generalizations of the classical development operation introduced by E. Cartan to define holonomy and which consists of rolling a Riemannian manifold *M* onto a tangent plane with no slipping nor spinning. In particular, we will consider the case of a Riemannian manifold *M*rolling onto a space form. We prove that the existence of a principal bundle connection associated to this rolling problem, which enables us to address controllability issues without any Lie bracket computations but instead by computing some holonomy groups. This is the joint work with M. Godoy Molina and P. Kokkonen.

Speaker: **Anton Thalmaier** (University of Luxembourg )

Title: **"Brownian motion with respect to evolving metrics and Perelman's entropy formula"**

Abstract: We discuss aspects of stochastic differential geometry in the case when the underlying manifold evolves along a geometric flow. Special interest lies in entropy formulas for positive solutions of the heat equation (or conjugate heat equation) under the Ricci flow.

March 27th

Speaker: **Stephan Wojtowytsch** (Master's student, ERASMUS)

Title: **"The Alexandrov topology in Sub-Lorentzian geometry"**

Abstract: ** **We will introduce the notion of the Alexandrov topology connected to the causal structure of spacetimes in Lorentzian geometry and general relativity, and deduce some of its properties. Then we investigate how it carries over to the more general sub-Lorentzian setting. Here due to the existence of singular curves, which we cannot use the calculus of variations on, the situation becomes more complex.

Time permitting we will touch upon the subject of length maximizing curves and the sub-Lorentzian time separation function. In all points we will try to contrast the phenomena present to the corresponding results from Riemannian and sub-Riemannian geometry.

March 20th

Speaker: **Georgy Ivanov** (PhD student, UiB)

Title: **"Loewner evolution driven by a stochastic boundary point"**

Abstract: The seminar is based on the paper G.Ivanov, A.Vasil'ev "Loewner evolution driven by a stochastic boundary point", Analysis and Mathematical Physics, 1:387--412, 2012. In that paper we use ideas of general Loewner theory to construct a class of processes having invariance properties similar to those of SLE.

February 28th

Speaker:** ****Alexey Tochin **(PhD student, UiB)

Title: **"Stochastic Loewner evolution and Conformal Field Theory "**

Abstract: We introduce basics of SLE (Stochastic Loewner evolution). One of the important problems in this theory is calculation of martingales as conservation (in mean) laws of this dynamical stochastic process. It turns out that this problem is related to well-known calculation of correlators in the Conformal Field Theory, in particular, to singular representations of the Virasoro algebra. We review these relations in a comprehensive way based on a series of papers by Bernard, Bauer, Werner, and Friedrich.

February 21st

Speaker:** ****Erlend Grong **(PhD student, UiB)

Title: **"****Stochastic Integration and stochastic differential equations with applications****"**

Abstract: The aim of the presentation is to give an introduction to the concept of random processes (also called stochastic processes), its integration and the notion of martingales. We look at the construction of the Itô integral and compare it to the construction of the integral with respect to a measure. We review some of the basic theorems and properties related to this. We end by discussing applications to stochastic differential equations (SDEs).

The talk is supposed to be understandable for audience that is not very familiar with the measure theory.

February 14th

Speaker:** ****Alexander Vasiliev **(Professor, UiB)

Title: **"Evolution of 2D-shapes"**

Abstract: The study of 2D-shapes is a central problem in computer vision.

Classification and recognition of objects from their observed silhouette (shape) is crucial. We give an overview of analysis of 2D-shapes via conformal welding and infinite-dimensional geometry.

January 31st, February 7th

Speaker: **Chengbo Li** (Tianjin University)

Title:** ****"Curvature invariants in contact sub-Riemannian structures and applications (I-II)"**

Abstract:** **We construct the curvature-type invariants of contact sub-Riemannian structures based on the study of differential geometry of curves in Lagrange Grassmannians in which we construct a complete system of symplectic invariants. The bridge between them is the so-called "Jacobi curves" associated with an extremal of the normal sub-Riemannian geodesic problem. The curvature invariants can be applied to the study of estimation of number of conjugate points of the extremal. If time permitting, we compare our construction of curvature invariants with that from the method of equivalence.

January 24th

Speaker: **Mahdi K. Salehani** (Postdoc, UiB)

Title: **"A geometric study of the three-body problem"**

Abstract: The "Newtonian three-body problem" is the mathematical study of how three heavenly bodies move in settings where the dynamics are dictated by Newton's law of motion. Like many mathematical problems, the simplicity of its statement belies the complexity of its solution. In fact, the problem has historically served as a source of mathematical discovery and new problems since 1687, the year of publication of Newton's "Principia mathematica."In this seminar, I shall present some results of my two recent works. Taking a differential geometric approach to the three-body problem -due to Wu-Yi Hsiang and Eldar Straume (2007, 2008), first a new family of periodic orbits for the planar three-body problem with non-uniform mass distributions will be exhibited. Then, applying an extension of Hamilton's principle to non-holonomic three-body systems, we obtain the generalized Euler-Lagrange equations of non-planar three-body motions; as an application of the derived dynamical equations, we answer the question raised by A. Wintner on classifying the "constant inclination solutions" of the three-body problem.

**2011**

2011, May 10th

Speaker: **Anastasia Frolova **(Master's student, UiB)

Title: **"Limit zero distributions of Heine-Stieltjes polynomials"**

Abstract: The seminar is based on the paper "Critical measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes polynomials" by A. Martínez-Finkelshtein, E. A. Rakhmanov. We consider Heine-Stieltjes polynomials - polynomial solutions of Lamé equation. We define extremal and critical measures in order to study limit zero distributions of such polynomials. We investigate connections of quadratic differentials with critical measures.

2011, May 3rd

Speaker: **Elena Belyaeva **(Master's student, UiB)

Title: **"Modulus method and its application to the theory of univalent functions"**

Abstract: We define a modulus of a family of curves according to the definition of Tamrazov and remind the notion of a quadratic differential on a Riemann surface. We consider the problem of defining a trajectory structure of quadratic differential depending on a parameter. Also, we consider one extremal problem which we solve using the modulus method.

2011, April 26th

Speaker: **Ksenia Lavrichenko** (Master's student, UiB)

Title: **"Moduli of system of measures on Heisenberg group"**

Abstract:** ** We shall define the p-module of system of measures according to the classical definition of B. Fuglede. We also recall our previous considerations of p-modulus of a family of curves joining the boundary components of the ring R in Heisenberg group. We explain the idea of a result of W. Ziemer and F. Gehring about relation of the conformal capacity of R to the extremal length of a family of surfaces that separate the boundary components of R in setting of Heisenberg group.

2011, April 12th

Speaker: **Georgy Ivanov** (PhD student, UiB)

Title: **"Loewner equation with moving boundary attractive point"**

Abstract:** **A general version of the Loewner equation has been developed since 2008 by Bracci, Contreras, Diaz-Madrigal and Gumenyuk. It was shown that there exists a 1-1 correspondence between Loewner type evolution families and Herglotz vector fields. We study the case when the attractive point of the Herglotz field moves along the boundary of the unit disc. In the deterministic case we let the point move with constant radial speed. In the stochastic case it realizes the Brownian motion on the circle.

2011, March 29th

Speaker: **Alexey Tochin** (PhD student, UiB)Title: **"Two mathematical problems from high energy physics and quantum field theory"**Abstract: We discuss two mathematically independent problems. The first one is related to meromorphic functions of two (or more) variables and their applications in relativistic quantum scattering theory. The condition of polynomial boundedness leads to a very hard restriction on the function parameters that can be compared with experimental data.The second part will be devoted to functional integral. One of the most physically important approaches to it is connected with a formal extension to perturbative series. It gives so-called Feynman graphs and Feynman rules, which play a critical role in high energy and elementary particle physics. The functional integral and the corresponding series admit invariants, that will also be a subject of our discussion..

2011, March 15th

Speaker: **Alexander Vasiliev** (Professor, UiB)

Title: **"Parametrization of the Loewner-Kufarev evolution in Sato's Grassmannian"**

Abstract: We discuss complex and Cauchy-Riemann structures of the Virasoro algebra and of the Virasoro-Bott group in relation with the Loewner-Kufarev evolution. Based on the Hamiltonian formulation of this evolution we obtain an infinite number of conserved quantities and provide embedding of the Loewner-Kufarev evolution into Sato's Grassmannian.

2011, March 8

Speaker: **Qifan Li **(Master's student, UiB)

Title: **"The Carleson-Hunt theorem"**

Abstract: The Carleson's famous paper in 1966 proved that the Fourier series of square-integrable functions converges almost everywhere. As indicated in Hunt's paper in 1967, Carleson's method can be modified to deal with the functions in L^{p}-space with p>1. In addition to Carleson's work, Fefferman provides another approach to solve this problem in 1971. His proof relies on the almost orthogonality property of the maximal Carleson operator on the time-frequency plane. This inspired the development of the theory of the time-frequency analysis. The joint paper of Lacey and Thiele in 2000 showed that the maximal Carleson operator can be decomposed in the time-frequency plane in terms of wave-packets and they provide a new proof of Carleson's theorem. We will follow the Carleson's approach in this talk and discuss the iteration arguments and the construction of exceptional sets.

2011, March 1

Speaker: **Erlend Grong **(PhD student, UiB)

Title: **"Infinite dimensional sub-Riemannian geometry"**

Abstract: We will talk about different attempts to study sub-Riemannian geometry in infinite dimensional manifolds.We will first to look at the development of the metric approach to study the space of shapes. Then I will talk about my recent work (joint work with Irina and Alexander), where I try to use the previous ideas in order to study the space of holomorphic functions. It turns out that many of the properties from sub-Riemannian geometry on finite dimensional principal bundles are generalized to this case.

2011, February 22

Speaker: **Christian Autenried **(Master's student, UiB)

Title: **"Universal Grassmannian (introduction and continuation)"**

Abstract: We shall define some dense submanifolds of the Universal Grassmannian and consider their properties. Then we shall study the stratification that gives us better understanding of the structure of the Grassmannian. The next step is to define the Pluecker coordinates and the embedding of the Grassmanian into projective space L2. Finally we shall see how the rotation group acts on Grassmannian and how this action is related to the stratification structure.

2011, February 15

Speaker: **Christian Autenried **(Master's student, UiB)

Title: **"Universal Grassmannian (introduction)"**

Abstract: This is the first lecture, where the definition of the infinite dimensional Grassmannian will be given. The simplest properties, such as manifold structure and action of the group will be considered.

2011, February 8

Speaker: **Irina Markina **(professor, UiB)

Title: **"Universal Grassmannian (introduction)"**

Abstract: In the following three lectures we will give the notion of an infinite dimensional analogue of the Grassmann manifold, that received the name Universal Grassmannian. In the first lecture I shall give auxiliary definitions from the functional analysis, such as the space of Hilbert-Schmidt and Fredholm operators, general linear restricted group and will provide elementary proofs and examples. In the following two lectures Christian Autenried will define the Universal Grassmannian as a manifold and present its properties.

**2010**

2010, November 23

Speaker: **Qifan Li** (Master's student, UiB)Title: "**The Q space and Triebel conjecture**"

Abstract: This talk is based on the paper http://arxiv.org/abs/0908.4380 which describes the Paley-Littlewood characterization of Q space and proved that Q space is exactly the space connecting the conjecture of Hans Triebel regarding an isomorphism theorem for elliptic operators in BMO space.

We refer to

Wen Yuan, Winfried Sickel and Dachun Yang, Morrey and Campanato meet Besov, Lizorkin and Triebel, Lecture Notes Math. 2005 (2010), Springer,

for the recent progress in this area.

2010, November 16

Speaker: **Ksenia Lavrichenko** (Master's student, UiB)

Title: **"Polar coordinates on Carnot groups"**Abstract: We describe a procedure for constructing "polar coordinates" in a certain classof Carnot groups elaborated by Z.M. Balog and J.T.Tyson (2002). We give explicit formulae for this construction in the setting of the Heisenberg group. The construction makes use of nonlinear potential theory, specially, the fundamental solutions to the p-sub-Laplace operators. One of the applications of this result is an exact capacity (module) estimate.Reference: Balogh, Zoltán M.(CH-BERN-IM); Tyson, Jeremy T.(1-SUNYS)Polar coordinates in Carnot groups. (English summary)Math. Z. 241 (2002), no. 4, 697-730.

2010, November 9

Speaker: **Anna Korolko** (PhD student, UiB)

Title:** "Variational Calculus**

**"**

2010, November 2

Speaker: **Georgy Ivanov** (PhD student, UiB)

Title:** "**

**Gaussian free field"**

Abstract**:** This seminar is an overview of the survey "Gaussian free field for mathematicians" by S. Scheffield (arXiv:math/0312099 [math.PR]). Gaussian free field (GFF), known in physics as the Euclidean bosonic massless free field, is an analog of Brownian motion for the case of d-dimensional time. It is an important object for many constructions in statistical physics. Due to its conformal invariance, the 2-dimensional GFF is a useful tool for studying Schramm-Loewner evolution (SLE).

2010, October 26

Speaker: **Georgy Ivanov** (PhD student, UiB)

Title**: "**

**Gaussian free field and conformal welding"**

Abstract**:** This seminar is based on the paper «Random curves by conformal welding» by K.Astala, P.Jones, A.Kupiainen, E.Saksman (2010). The authors construct a conformally invariant random family of Jordan curves in the plane by welding random homeomorphisms of the unit circle generated by the exponential of the trace of the 2-dimensional Gaussian Free Field (GFF). This construction is in a certain sense an analog of Schramm-Loewner Evolution (SLE) for the case of closed curves.

We will start by giving the definitions of the trace of GFF and of the problem of conformal welding. Then we will give an outline of the construction and, if time permits, some of the technical details.

2010, October 19

Speaker: **Mauricio Godoy** (PhD student, UiB)

Title:** "**

**On Gromov's theorem on group growth"**

Abstract: One of the most celebrated results of M. Gromov is the characterization of finitely generated groups of polynomial growth as groups with a nilpotent subgroup of finite index, called almost nilpotent groups. The proof and some related results of this theorem have a strong "sub-Riemannian flavour". For example, the degree of the polynomial growth, given by the Bass-Milnor-Wolf formula, also known as Bass-Guivarch formula, is surprisingly similar to Mitchell's formula for the Hausdorff dimension of a sub-Riemannian manifold at a regular point.

The aim of this talk is to present the necessary definitions, to sketch the proof of Gromov's theorem from a sub-Riemannian point of view and to study some examples.

2010, October 12

Speaker: **Anastasia Frolova** (Master's student, UiB)

Title: "**Modifications of the Schwarz lemma for regular functions on a free domain**"

Abstract: We consider regular functions defined on an arbitrary open subset of the unit disk containing zero. Using classical properties of conformal maps and a sufficient condition for univalence for such functions we get special modifications of the Schwarz lemma and inequalities for the functions’ coefficients. In particular, we apply such estimates to algebraic polynomials.

2010, October 5

Speaker: **Erlend Grong **(PhD student, UiB)

Title**:** **"Rolling and controllability"**

Abstract**:** Earlier this year, me and some friends (Mauricio, Irina and Fatima) submitted a paper where we have been working on an intrinsic formulation of the problem of rolling manifolds. Our work was inspired by ideas of Agrachev and Sachkov on two dimensional manifolds. Their result about controllability explains how the Gaussian curvature determines which points on the manifold can be reached, given an initial configuration of two rolling bodies. I will present now the notion that acts like an analogue for controllability condition in higher dimensions when one works with rolling problem.

2010, September 28

Speaker: **Mauricio Godoy** (PhD student, UiB)

Title: **"On the Rumin-Ge complex"**Abstract: On the famous survey "Carnot-Carathéodory spaces seen from within", Mikhail Gromov proposes, among other ideas, a theory of horizontal differential forms for contact manifolds. This approach was subsequently explored by Michel Rumin, and extended (in a "non-canonical" way) to more general classes of sub-Riemannian manifolds by Zhong Ge. In this seminar I will present some of the basic constructions and ideas behind the theory and we will see how these give a natural environment for a Hodge theory on sufficiently nice sub-Riemannian manifolds.

2010, September 21

Speaker: **Takaharu Yaguchi** (The University of Tokyo)

Title: **The Discrete Variational Derivative Method Based on Discrete Differential Forms**

Abstract: As is well known, for PDEs that enjoy a conservation or dissipation property, numerical schemes that inherit the property are often advantageous in that the schemes are fairly stable and give qualitatively better numerical solutions in practice.Lately, Furihata and Matsuo have developed the so-called “discrete variational derivative method” that automatically constructs energy preserving or dissipative finite difference schemes. Although this method was originally developed on uniform meshes, the use of non-uniform meshes is of importance for multi-dimensional problems.In this talk, we will show an extension of this method to triangular meshes. This extension is achieved by combination of this method and the theory of the discrete differential forms by Bochev and Hyman.

2010, September 15

Speaker: **Marek Grochowski** (Cardinal Stefan Wyszynski University in Warsaw) Title: "**An 'algorithm' for computing reachable sets for some sub-Lorentzian structures on R ^{3}**"Abstract: The aim of my talk is to show a kind of algorithm allowing to construct functions defining reachable sets for certain sub-Lorentzian structures on R

^{3}, including contact and Martinet sub-Lorentzian structures. A number of functions (which can be equal to 2 or 4) needed for describing the (future) nonspacelike reachable set from a point q depends on whether there exists or there does not exist a timelike abnormal curve contained in the boundary of the reachable set from q.

2010, September 3

Speaker: **Anna Korolko **(PhD student, UiB)

Title: "**Sub-Lorentzian geometry**"

Abstract: My task would be to explain you basic facts and main definitions from sub-Lorentzian geometry.

2010, August 31

Speaker: Professor **Alexander Vasiliev** (UiB)

Title: "**Tangential properties of trajectories for holomorphic dynamics in the unit disk**"

Abstract: We consider dynamics of holomorphic selfmaps of the unit disk with a Denjoy-Wolff (DW) point of hyperbolic type at the boundary. Contreras and Diaz-Madrigal proved that if two dynamics have the same DW point such that any point of the unit disk approaches after iterations the DW point with the same tangent line at DW, then they are the same. Bracci supposed that we need to check this property only at finite number of points in the unit disk. We disprove this conjecture.

2010, August 25

Speaker: Professor **Alexander Olevskii **(Tel Aviv University, Israel)

Title: "**Wiener's "closer of translates" problem**"

Abstract: Wiener characterized cyclic vectors (with respect to translations) in L_{1} (R) and L_{2}(R) in terms of zero sets of Fourier transform. He conjectured that similar characterization should be true for L_{p} (R) , 1<p

** **

2010, June 25

Speaker: Professor **Vladislav Poplavskii** (Saratov State University)

Title: "**On determinants of Boolean matrices**"

Abstract: We introduce the notion of the determinant of the square matrices over Boolean algebra. We present applications of the determinant are under consideration to the theory of rank functions and to solution of linear systems of inequalities and equations.

2010, June 9

Speaker: Professor** Dmitri Prokhorov** (Saratov State University, Russia)

Title: "**Integrability cases of the Loewner equation**"

Abstract: We give the partial cases of the Loewner equation which can be integrated in quadratures. The corresponding mapping properties are described.

2010, June 1

Speaker: Professor** Martin Schlichenmaier** (University of Luxembourg)

Title: **"Krichever - Novikov type algebras - an overview**"

Abstract: The Witt algebra, its central extension the Virasoro algebra, and the affine Lie algebras play an important role in a number of fields. From the geometric point of view they are infinite-dimensional Lie algebras of meromorphic objects assocated to the Riemann sphere. Coming from applications there is a need for similar constructions for higher genus Riemann surfaces. They are given by Krichever - Novikov type algebras. In this talk we will introduce them and discuss their properties.

2010, May 18

Speaker: **Anna Korolko** (PhD student, UiB)

Topic: "**Differential equations in Matrices and Matrix exponential**"

Abstract: The exponential function for matrices will be introduced and one-parameter subgroups of matrix groups will be studied. We will show how these ideas can be used in the solution of certain types of differential equations.

2010, April 27

Speaker: **Qifan Li** (Master's student UiB)

Title: "**Proof of Astala's conjecture**" *continued*

Abstract: Quifan will continue and present the proof of a couple lemmas that he used in the proof of the main theorem.

2010, April 20

Speaker: **Qifan Li** (Master's student UiB)

Title: "**Proof of Astala's conjecture**"

Abstract: We will discuss the proof of Astala's conjecture which is the ground-breaking work of Michael T. Lacey, Eric T. Sawyer and Ignacio Uriarte-Tuero. The new idea in the paper is the proof of boundedness of a certain Calderon-Zygmund operators on spaces with non-doubling weights which will be also discussed in the presentation.

Reference: Michael T. Lacey, Eric T. Sawyer, Ignacio Uriarte-Tuero: Astala's Conjecture on Distortion of Hausdorff Measures under Quasiconformal Maps in the Plane. arXiv:0805.4711v3 [math.CV]

2010, April 13

Speaker: Ksenia Lavrichenko (Master student UiB)

Title: "**Liebermann theorem for a particular case of Heisenberg group**"

Abstract: We consider contact transformations on three-dimensional Heisenberg group. It is well known that, for example, group SU(1,2) belongs to the class of contact transformations on Heisenberg group. The question arises are there any more? We shell discuss the theorem that gives the conditions under which one can produce the contact map flows by vector fields of a special form.

Reference: A.Koranyi, H.M.Reimann "Quasiconformal mappings on the Heisenberg group", 1985.

2010, April 6

Speaker: **Elena Belyaeva** (Master's student UiB)

Title: "**Quadratic differentials on a Riemann surface**"

Abstract: A quadratic differential on a Riemann surface is locally represented by a meromorphic function that changes by means of multiplication by the square of the derivative under a conformal change of the parameter. It defines, in a natural way, a field of line elements on the surface, with singularities at the critical points of the differential, i. e. its zeros and poles. The integral curves of this field are called the trajectories of the differential. We consider the local a global trajectory structure of quadratic differentials and define completely the structure of trajectories in a special case.

2010, March 23

Speaker: **Vendula Exnerova**

Title: "**Bifurcation along a non-degenerated eigenvalue**"

Abstract: After an introduction to the bifurcation basic terms I would like to go on through Lyapunov-Schmidt reduction. With some preparation I would like to prove the Theorem about bifurcation along non-degenerated eigenvalue.

2010, February 23Speaker: Professor **Alexander Vasiliev** (UiB)Title: "**Quantum harmonic oscillator and the Bloch sphere**" *continued*

Abstract: We shall discuss some quantum mechanics underlying the Heisenberg uncertainty and the Hopf principle bundle. We start with the simplest quantum harmonic oscillator. The symmetry is given by the energy conservation law. Then we turn to a closed system of N interacting particles with symmetries given my the angular momentum conservation. We shall discuss similarities and differences in these two models.

2010, February 16Speaker: Professor **Alexander Vasiliev** (UiB)Title: "**Quantum harmonic oscillator and the Bloch sphere**"Abstract: We shall discuss some quantum mechanics underlying the Heisenberg uncertainty and the Hopf principle bundle. We start with the simplest quantum harmonic oscillator. The symmetry is given by the energy conservation law. Then we turn to a closed system of N interacting particles with symmetries given my the angular momentum conservation. We shall discuss similarities and differences in these two models.

2010, February 9Speaker: Professor **Irina Markina** (UiB)Title: "** ****The Virasoro group as a complex manifold**" *continued*The speaker will remined definitions and prove that the group *Diff* has CR-structure.

2010, January 26Speaker: Professor **Irina Markina** (UiB)Title: "**The Virasoro group as a complex manifold**"Abstract: The main purpose of the talk is to discuss the geometric structure of the group *Diff* of sense preserving diffeomorphisms of the unite circle *S*. It appears that it is an infinitedimensional CR-manifold in some complex Frechet space. A shall provide all necessary definitions. The Virasoro group *Vir* is a central extension of *Diff* by real numbers. We will see that the map *Vir* to* Diff/S* is a holomorphicaly trivial principal C*-bundle.

## 2009

2009, November 24

Speaker: **Henning Abbedissen Alsake****r** (Master's student, UiB)

Title: "**Multipliers of the Dirichlet space**"

Abstract: We define and study the Dirichlet space and some related spaces of analytic functions. We then address the problem of characterizing the multipliers of these spaces. Finally, if time allows, we consider the multipliers as a Banach algebra and state some results and pose some questions in this direction.

2009, November 17Speaker: PhD student **Anna Korolko** (UiB) Title: "**Generalized Heisenberg Groups**"

Abstract: We will discuss two-step nilpotent Lie groups with a natural left-invariant metric and consider some of their geometry. These groups constitute a natural generalization of the Heisenberg group.

2009, November 10Speaker: **Georgy Ivanov** (Master's student, UiB)Title: "**One-slit dynamics of domains and the norms of a driving term in the Loewner-Kufarev equation**"Abstract: It has been known since 1923 that every single-slit mapping which satisfies certain normalization conditions can be represented as a solution of the Loewner equation with an appropriately chosen driving term, which is a continuous real-valued function.In 1947 Kufarev gave an example showing that the converse is not true, i.e., there exists a continuous driving term which generates a non-slit mapping. He also found a sufficient condition for a driving term to generate a one-slit mapping, namely the boundedness of the driving term’s first derivative.The second known sufficient condition was given by Marshall, Rohde and Lind in 2005. They showed that if Lip(1/2)-norm of the driving term is less than 4, then the Loewner equation will generate a slit map.

We construct a family of examples of non-slit solutions which includes Kufarev’s example as a trivial case. This family contains both examples where the Lip(1/2)-norms are arbitrarily large and where they approach 4 from above arbitrarily close.

2009, November 3Speaker: Postdoc **Pavel Gumenyuk** (UiB)Title: "**Geometry behind Loewner chains**"

Abstract: This talk is a continuation of the previous seminar held by Prof. Santiago Díaz-Madrigal on Tuesday last week. There will be presented recent results on the admissible geometry for Loewner chains of chordal type in the most general case as well as in the special considered by V.V. Goryainov and I.Ba (1992) and by R.O.Bauer (2005). These results are achieved in collaboration with Prof. Manuel D. Contreras and Santiago Díaz-Madrigal from the University of Sevilla, SPAIN.

2009, November 2

Speaker: Dr. **Yu-Lin Lin** (Institute of Mathematics, Academia Sinica, Taipei, Taiwan)

Title: "**Large-time rescaling behaviors to the Hele-Shaw problem driven by injection**"

Abstract: This talk addresses a large-time rescaling behavior of Hele-Shaw cells for large data initial domains. The Polubarinova-Galin equation is the reformulation of zero surface tension Hele-Shaw flows with injection at the origin in two dimensions by considering the moving domain $\Omega(t)=f(B_{1}(0),t)$ for some Riemann mapping f(z,t). We give a sharp large-time rescaling behavior of global strong polynomial solutions to this equation and the corresponding moving boundary in terms of the invariant complex moments. Furthermore, by proving a perturbation theorem of polynomial solutions, we also show that a small perturbation of the initial function of a global strong polynomial solution also gives rise to global strong solution and a large-time rescaling behavior of the moving domain is shown as well.

2009, October 27

Speaker: Professor **Santiago Díaz-Madrigal** (joint work with professor **Manuel Contreras**), University of Seville

Title: "**Generalized Loewner theory in the unit disk**"

Abstract: We introduce a general version of the notion of Loewner chains and Loewner differential equations which extend and unify the classical cases of the radial and chordal variant of the Loewner differential equation as well as the theory of semigroups of analytic functions. In this very general setting, we establish a deep correspondence between these chains and the weak solutions of some specific non-autonomous differential equations. Among other things, we show that, up to a Riemman map, such a correspondence is one-to-one. In a similar way as in the classical Loewner theory, we prove that these chains are also solutions of a certain partial differential equation which resembles (and includes as a very particular case) the classical Loewner - Kufarev PDE.

2009, October 20

Speaker: PhD student **Mauricio Godoy Molina** (UiB) Title: **"Looking for (sR)geodesics and (sR)Laplacians on spheres" **

Abstract: In this talk I will present some of our attempts for finding "convenient" distributions on odd dimensional spheres, and some consequences of their existence. Our primary goals are describing the sub-Riemannian geodesics and the intrinsic sub-Riemannian Laplacian induced by these distributions. A more important goal (but considerably harder) is finding the sub-Riemannian heat kernel, which will eventually lead to a closed expression for the associated Carnot-Carathéodory distance. This last part promises to be sketchy and incomplete, but motivational.

2009, October 13

Speaker: Professor **Irina Markina** (UiB)

Title: **"Quasiconformal mapping on the Heisenberg group" (continuation)**

Abstract: In the first part of the talk we showed how the one-dimensional Heisenberg group appeared in the Bruhat decomposition of the group SU(1,2). The second part will be devoted to the definitions of contact and quasiconformal mappings on the Heisenberg group. After formulating some properties of quasiconformal mappings we prove the existance of a flow of contact maps on the Heisenberg group.

2009, October 6

Speaker: Professor **Irina Markina** (UiB)

Title: **"Quasiconformal mapping on the Heisenberg group"**

Abstract: In the first part of the talk we show how the one-dimensional Heisenberg group appears in the Bruhat decomposition of the group SU(1,2). The second part will be devoted to the contact and quasiconformal mappings on the Heisenberg group. After formulating some properties of quasiconformal mappings we prove the existance of a flow of contact maps on the Heisenberg group.

2009, September 29

Speaker: Professor **Alexander Vasiliev** (UiB)

The 2nd talk in the mini-course

"**Quantum underdamped dissipative harmonic oscillator**"

2009, September 22

Speaker: Professor **Alexander Vasiliev** (UiB)

Title: "**Quantum underdamped dissipative harmonic oscillator**"

Abstract: We give some basics of quantum mechanics arriving at classical and quantum harmonic oscillator. We shall analyze a simplest example of mixed divergent-curl system, i.e., an underdamped dissipative harmonic oscillator, and present its first quantization using complex form of the Hamiltonian.

2009, September 15

Speaker: master student **Elena Belyaeva** (UiB)

Title: "**Nash equilibrium in games with ordered outcomes**"

Abstract:

This work is devoted to one special sort of games, studied with theory of games. A subject matter of this theory is situations where several sides participate, and every of sides pursues its own goal. The result, or final state of situation, is defined with joint actions of all sides. These situations are called games.

Theory of games explore the possibilities of colliding sides and attempts to define such strategy for every player that the result of the whole game would be best in certain sense, called principle of optimality (we consider Nash principle of optimality).

The main aim of current work is finding criterion conditions for existing a Nash equilibrium situations in mixed expansion of game with ordered outcomes. In part I we set a connection between Nash equilibrium situations and balanced submatrixes of payoff function’s matrix. In part II we found required and sufficient conditions for balanced matrix. In appendix there is a program for finding a Nash equilibrium situations in arbitrary finite game of two players with ordered outcomes.

2009, September 8

Speaker: master student **Ksenia Lavrichenko** (UiB)

Title: **"Investigation of phase portraits of three-dimensional models of gene networks"**

Abstract.

- Motivation: Prediction of regimes of molecular-genetic system functioning by structural and functional organization of a system is one of the key problems in the fields of bioinformatics studying gene network functioning. To address this problem, it is necessary to perform theoretical studies of functioning of gene networks’ regulatory contours and to reveal their general regularities, which determine the presence or absence of ability to support stationary, cyclic, or other, more complex regimes of functioning.
- Results: Presence and stability of the limit cycles and stationary points of small amplitude resulting from the Andronov–Hopf bifurcation were studied in a system of ordinary differential equations which describes the behavior of a three-dimensional hypothetical gene regulatory network.

2009, September 1Speaker: Ph.D. student **Erlend Grong**, University of Bergen, NorwayTitle: "**Sub-Riemannian and sub-Lorentzian geometry on SU(1,1) and its universal covering**"

Abstract: We discuss the example of SU(1,1) with the pseudometric induced by the Killing form. Choosing different types of distributions, we get a sub-Riemannian and a sub-Lorentzian manifolds. We also lift these structures to the universal cover CSU(1,1). In the sub-Riemannian case, we find the distance function and describe the number of geodesics on SU(1,1) and CSU(1,1) completely. This is example is important because, unlike the Heisenberg group, the cut and conjugate loci do not coincide. Furthermore, we describe the sub-Lorentzian geodesics and compare them to the Lorentzian ones. This example is important because the CSU(1,1) with the induced Lorentzian metric is isometric to the anti-de Sitter space (or the universal cover of it, depending on how you define it).

2009, August 25

Speaker: Professor** David Shoikhet**,** **Department of Mathematics, ORT Braude College, Karmiel, Israel

Title: "**A flower structure of backward flow invariance domains**"

Abstract: We study conditions which ensure the existence of backward flow invariant domains for semigroups of holomorphic self-mappings of a simply connected domain $D$. More precisely, the problem is the following. Given a one-parameter semigroup $S$ on $D$, find a simply connected subset $\Omega\subset D$ such that each element of $S$ is an automorphism of $\Omega$, in other words, such that $S$ forms a one-parameter group on $\Omega$.

2009, June 23

Speaker: **Fátima Silva Leite**,Department of Mathematics and Institute of Systems and Robotics, University of Coimbra, Portugal

Title: **"The geometry of rolling maps"**

Abstract: Rolling maps describe how one smooth manifold rolls on another, without twist or slip. We will focus on the geometry of rolling a Riemannian manifold on its affine tangent space at a point. Both manifolds are considered to be equipped with the metric induced by the Euclidean metric of some embedding space.

The Kinematic equations of a rolling motion can be described by a control system with constraints on velocities, evolving on a subgroup of the Euclidean group of rigid motions, describing simultaneously rotations and translations in space. Choosing the controls is equivalent to choosing one of the curves along which the two manifolds touch. Issues like controllability and optimal control of rolling motions will be addressed and illustrated for the most well studied of these nonholonomic mechanical systems, the rolling sphere.

Other interesting geometric features of rolling motions will be highlighted.

2009, May 12

Speaker: **Alexander Vasiliev**, University of Bergen, Norway

Title: "**Quantization of dissipative systems and complex Hamiltonians**"

Abstract: We start with the classical notion of the first quantization and give the Dirac scheme using ladder operators. Then we suggest a general approach to quantization of dissipative systems, in which the imaginary part of the complex Hamiltonian plays the role of entropy. The damped harmonic oscillator is considered as a typical example.

2009, May 5

Speaker: **Irina Markina**, University of Bergen, Norway

Title: "**Why is sub-Riemannian geometry applicable?**"

Abstract: A sub-Riemannian geometry of 3D sphere can be defined by means of the Hopf fibration. We will give all necessary definitions, and describe a sub-Riemannian structure on the 3D sphere using the Hopf map and Ehresmann connection. Then we describe states and state vectors of the two-level quantum systems (qubits) and show how they lead to the Hopf map. At the end, we discuss adiabatic transport of the state vectors over curves in the Bloch sphere, that are sub-Riemannian geodesics in the geometric language.

2009, April 28

Speaker: **Arne Stray**, University of Bergen, Norway

Title: "**Extremal solutions to the Nevanlinna-Pick problem**"

2009, April 21

Speaker: **Henrik Kalisch**, University of Bergen, Norway

Title: "**Non-existence of solitary water waves in three dimensions**"

Abstract: This talk will be about a paper of Walter Craig, concerning nonexistence of localized solitary-wave solutions in three dimensions.

References: MR1949966 (2003m:76011) Craig, Walter Non-existence of solitary water waves in three dimensions. Recent developments in the mathematical theory of water waves (Oberwolfach, 2001). R. Soc. Lond. Philos. Trans. Ser. A Math. Phys. Eng. Sci. 360 (2002), no. 1799, 2127--2135. (Reviewer: Nikolay G. Kuznetsov) 76B03 (35J65 35Q51 35Q53 76B15 76B25)

2009, March 31

Speaker: **Mauricio Godoy Molina**, University of Bergen, Norway

Title: "**Sub-Riemannian geodesics of odd-dimensional spheres**"

Abstract: In this short talk, two interesting results will be presented; one concerning normal sub-Riemannian geodesics when the manifold is a principal G-bundle (for a suitable G) and the other concerning the construction of Popp's measure for odd-dimensional spheres. The first theorem will be applied to determine all possible normal (and thus all) sub-Riemannian geodesics when G=S1 and G=S3, and the second one will be applied in determining the intrinsic hypoelliptic Laplacian for S7, when the horizontal distribution has rank 6.

## 2008

Tuesday, December 9, 2008, Aud. Pi, 14:15Speaker: **Roland Friedrich** (Max-Planck-Institut für Mathematik, Bonn, Germany)"Aspects of the Global Geometry underlying Stochastic Loewner Evolutions"

Tuesday, December 2, 2008, room. 526, 14:15Speaker: **Pavel Gumenyuk** (UiB)"Loewner chains in the unit disk"

Tuesday, November 18, 2008, room. 526, 14:15Speaker: **Mauricio Godoy** (UiB)"Global Sub-Riemannian Geometry of Spheres"

Tuesday, November 11, 2008, room. 526, 14:15Speaker: **Anna Korolko** (UiB)"Sub-semi-Riemannian geometry"

Tuesday, October 28, November 4, 2008, room. 526, 14:15Speaker: **Erlend Grong** (UiB)"Optimal control and geodesics on anti-de Sitter space"

Tuesday, October 21, 2008, room. 640, 14:15Speaker: **Dante Kalise** (UiB)"Numerical approximation of an optimal control problem in a stronglydamped wave equation"

Tuesday, October 14, 2008, room. 526, 14:15Speaker: **Georgy Ivanov** (UiB)"Martingales with applications to Brownian motion and Walsh series"

Tuesday, September 30, October 7, 2008, room. 526, 14:15Speaker: **Irina Markina** (UiB)"Rashevskii theorem"

Tuesday, September 23, 2008, room. 526, 14:15Speaker: **Alexander Vasiliev** (UiB)"Slit-solutions to the Loewner-Kufarev equation"

Tuesday, April 15, 2008, room. 534, 15:00Speaker: **Peter A. Clarkson** (Kent University, UK)"Rational solutions of soliton equations"

Tuesday, January 29, 2008, room. 534, 14:15Speaker: **Anna Korolko** (UiB)"The pointwise inequalities for Sobolev functions on Carnot groups"

Tuesday, January 15, 2008, room. 534, 14:15Speaker: **Alexander Vasiliev** (UiB)"Virasoro Algebra and Loewner Chains"

## 2007

Tuesday, October 23, 2007, room. 534, 14:15Speaker: **Alexander Vasiliev** (UiB)"From Hele-Shaw flows to Integrable Systems. Historical Overview"

Tuesday, October 2,9, 2007, room. 534Speaker: **Irina Markina** (UiB)"Rotations, unit S^3 sphere, and Hopf fibration"

Joint seminar (Analysis and Image Procesing Groups)Tuesday, September 18, 2007, room. 534Speaker: **Dominque Manchon** (Blaise Pascal University, France)"Dendriform algebras and a pre-Lie Magnus type expansion"(joint work with Kurusch Ebrahimi-Fard)

Tuesday, September 11, 2007, room. 534Speaker: **Arne Stray** (UiB)"Restrictions of the disc algebra described locally"

Tuesday, September 4, 2007, room. 534Speaker: **Pavel Gumenyuk** (UiB, Norway; Saratov State University, Russia)"Siegel disks and basins of attraction"

Tuesday, April 24, 2007, room. 526Speaker: **Erlend Grong** (UiB)"On the distortion of the conformal radius under quasiconformal map"

Wednesday, April 18, 2007, aud. "Pi"Speaker: **Semen Nasyrov** (Kazan State University, Russia)"Lavrentiev problem for an airfoil"

Wednesday, March 14, 2007, aud. "Pi"Speaker: **Yurii Semenov** (NTNU, Trondheim)"Complex variables in the water entry problem"

Wednesday, February 7, 2007, aud. "Pi"

Wednesday, February 14, 2007, aud. "Pi" (continuation)

Wednesday, February 21, 2007, aud. "Pi" (final part)Speaker: **Alexander Vasiliev** (UiB)"Virasoro Algebra: Analysis, Geometry, Integrability"

## 2006

Thursday, September 28, 2006, room 510Thursday, October 19, 2006, room 534 (continuation)Speaker: **Irina Markina** (UiB)"Some interesting examples of Heisenberg-type homogeneous groups"

Wednesday, September 13, 2006, Auditorium PiJoint Seminar of Pure Mathematics GroupsSpeaker: **Rubén Hidalgo** (Universidad Técnica Federico Santa María, Valparaíso, Chile)"Extended Schottky groups"

Wednesday, September 6, 2006, room 508Speaker: **Alexander Vasiliev **(UiB)"Lower Schwarz-Pick estimates and angular derivatives"Wednesday, August 16, 2006, Auditorium Pi

Analysis Seminar and Department's ColloquiumSpeaker: **Dmitri Prokhorov** (Saratov State University, Russia)"Dynamical systems and the Loewner equation"

Wednesday, May 10, 2006, Auditorium PiAnalysis Seminar and Department's ColloquiumSpeaker: **J. Milne Anderson** (University College London, UK)"Cauchy transform of point masses"

Wednesday, April 26, 2006, room 526Speaker:** Yurii Lyubarskii **(NTNU, Trondheim)"On decay of holomorphic functions"

Wednesday, March 29, 2006, room 526Speaker: **Irina Markina** (UiB)Title: "About Heisenberg group" (final talk)

Wednesday, March 22, 2006, room 526Speaker: **Irina Markina** (UiB)Title: "About Heisenberg group" (continuation)

Wednesday, March 15, 2006, auditorium PiJoint Analysis seminar and Department's colloquiumSpeaker: **Björn Gustafsson** (KTH, Stockholm)Title: "On inverse balayage and potential theoretic skeletons"

Wednesday, March 8, 2006, room 526Speaker: **Irina Markina** (UiB)Title: "About Heisenberg group"

Thursday, February 2 and 16, 2006, room 510Speaker: **Alexander Vasiliev** (UiB)Title: "Bosonic strings and subordination evolution"

## 2005

Thursday, December 8, 2005, room 526Speaker: **Arne Stray** (UiB)Title: "A problem about harmonic functions"

Thursday, October 13, 2005, room 526Speaker: **Alexander Vasil'ev** (UiB)Title: "Modeling 2-D flows in Hele-Shaw cells by conformal maps"

Thursday, September 22, 2005, room 526Speaker: **Giuseppe Coclite** (CMA Oslo and University of Bari, Italy)Title: "Global Weak Solutions to a Generalized Hyperelastic-Rod Wave Equation"