# Archive of seminars 2016-2020

Speakers at the Analysis and PDE seminar from earlier semesters in 2016 to 2020.

## Main content

**2020 (Fall)**

-- Break due to corona lockdown ---

**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. Inthe 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 theFourier restriction norm method in the framework of U^2-, V^2-spacesto prove the small data global well-posedness and scattering of theDysthe 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 27: Break due to workshop "Analysis and Geometry in Norway" on Friday October 30.

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

Contact the seminar organizer for link.

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

**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 13: Break due to to session of MAT331: Infinite-dimensional geometry.

**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 15: Break due to to session of MAT331: Infinite-dimensional geometry.

**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).

**2020 (Spring)**

The Analysis and PDE seminar is cancelled for the rest of the semester. See you in the fall of 2020.

**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 (Fall)**

**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.

**2019 (Spring)**

**Day:** February 12, 2019

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

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

**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 26, 2019

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

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

**Day:** April 02, 2019

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

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

**Day:** April 30, 2019

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

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

**Day:** May 07, 2019

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

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

**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**

**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**

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**

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.