# Analysis and PDE Seminar

Analysis and PDE seminar takes place weekly, on Tuesdays, at 14.15. Address: seminar room SP2 (2018), Realfagbygget, Allégaten 41.

## NEXT SEMINAR:

**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

## FOTHCOMING SEMINARS:

**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:** Mai 08, 2018

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

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

**Title:**

**Abstract:**

**RECENT SEMINARS:**

**2018**

**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.

**ARCHIVE**

**2017**

**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.

**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 Professor

Department 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, and

Department 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 free

boundary in the thin obstacle problem.**Abstract**: Monotonicity formulas play a pervasive role in the study of

variational inequalities and free boundary problems. In this talk we

will describe a new approach to a classical problem, namely the thin

obstacle (or

Signorini) problem, based on monotonicity properties for a family of

so-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. They

fairly desicribe wave propagation and are subject of a strong

interest. While the global construction of a singe nodal set

hardly can be well understood, one may hope that common nodal

sets of large families of eigenfunctions must have a pretty

special geometry. In particular, it was conjectured that

common nodal hypersurfaces of eigenfunctions,arising as the

spectrum of a compactly supported function, are

cones-translates of the zero sets of nonzero harmonic

homogeneous polynomials (spatial harmonics). It is confirmed

in 2d and is still open in higher dimensions. The approaches

and 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 as

a background tool for some integrable systems. In particular,

splitting time leads to the Vlasov equation for the distribution

function of plasma, and to the Benney hierarchy. We also show that

the solution to the Loewner equation with infinite dimensional time gives

the Lax function which solves the dispersionless KP hierarcy in

which the Benney equations may be recovered as the second in

the 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 on

geometric properties of a slit. For a domain $\Omega$ slit along a curve $\gamma=\gamma[0,t]$ and

for 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 sides

of $\gamma[0,t]$. We estimate asymptotically $$\frac{m_1(t)}{m_2(t)},\;\;\;t\to+0,$$ taking into

account 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 into

non-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 estimates

for 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 Kang

http://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 class

of 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 23

Speaker: 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 16

Speaker: 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 9

Speaker: 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 26

Speaker: 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 17

Speaker: 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 10

Speaker: **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 3

Speaker: 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 1

Speaker: Ph.D. student **Erlend Grong**, University of Bergen, Norway

Title: "**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, 14:15

Speaker: **Roland Friedrich** (Max-Planck-Institut für Mathematik, Bonn, Germany)

"Aspects of the Global Geometry underlying Stochastic Loewner Evolutions"

Tuesday, December 2, 2008, room. 526, 14:15

Speaker: **Pavel Gumenyuk** (UiB)

"Loewner chains in the unit disk"

Tuesday, November 18, 2008, room. 526, 14:15

Speaker: **Mauricio Godoy** (UiB)

"Global Sub-Riemannian Geometry of Spheres "

Tuesday, November 11, 2008, room. 526, 14:15

Speaker: **Anna Korolko** (UiB)

"Sub-semi-Riemannian geometry"

Tuesday, October 28, November 4, 2008, room. 526, 14:15

Speaker: **Erlend Grong** (UiB)

"Optimal control and geodesics on anti-de Sitter space"

Tuesday, October 21, 2008, room. 640, 14:15

Speaker: **Dante Kalise** (UiB)

"Numerical approximation of an optimal control problem in a strongly

damped wave equation"

Tuesday, October 14, 2008, room. 526, 14:15

Speaker: **Georgy Ivanov** (UiB)

"Martingales with applications to Brownian motion and Walsh series"

Tuesday, September 30, October 7, 2008, room. 526, 14:15

Speaker: **Irina Markina** (UiB)

"Rashevskii theorem"

Tuesday, September 23, 2008, room. 526, 14:15

Speaker: **Alexander Vasiliev** (UiB)

"Slit-solutions to the Loewner-Kufarev equation"

Tuesday, April 15, 2008, room. 534, 15:00

Speaker: **Peter A. Clarkson** (Kent University, UK)

"Rational solutions of soliton equations"

Tuesday, January 29, 2008, room. 534, 14:15

Speaker: **Anna Korolko** (UiB)

"The pointwise inequalities for Sobolev functions on Carnot groups"

Tuesday, January 15, 2008, room. 534, 14:15

Speaker: **Alexander Vasiliev** (UiB)

"Virasoro Algebra and Loewner Chains"

### 2007

Tuesday, October 23, 2007, room. 534, 14:15

Speaker: **Alexander Vasiliev** (UiB)

"From Hele-Shaw flows to Integrable Systems. Historical Overview"

Tuesday, October 2,9, 2007, room. 534

Speaker: **Irina Markina** (UiB)

"Rotations, unit S^3 sphere, and Hopf fibration"

Joint seminar (Analysis and Image Procesing Groups)

Tuesday, September 18, 2007, room. 534

Speaker: **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. 534

Speaker: **Arne Stray** (UiB)

"Restrictions of the disc algebra described locally"

Tuesday, September 4, 2007, room. 534

Speaker: **Pavel Gumenyuk** (UiB, Norway; Saratov State University, Russia)

"Siegel disks and basins of attraction"

Tuesday, April 24, 2007, room. 526

Speaker: **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 510

Thursday, October 19, 2006, room 534 (continuation)

Speaker: **Irina Markina** (UiB)

"Some interesting examples of Heisenberg-type homogeneous groups"

Wednesday, September 13, 2006, Auditorium Pi

Joint Seminar of Pure Mathematics Groups

Speaker: **Rubén Hidalgo** (Universidad Técnica Federico Santa María, Valparaíso, Chile)

"Extended Schottky groups"

Wednesday, September 6, 2006, room 508

Speaker: **Alexander Vasiliev **(UiB)

"Lower Schwarz-Pick estimates and angular derivatives"

Wednesday, August 16, 2006, Auditorium Pi

Analysis Seminar and Department's Colloquium

Speaker: **Dmitri Prokhorov** (Saratov State University, Russia)

"Dynamical systems and the Loewner equation"

Wednesday, May 10, 2006, Auditorium Pi

Analysis Seminar and Department's Colloquium

Speaker: **J. Milne Anderson** (University College London, UK)

"Cauchy transform of point masses"

Wednesday, April 26, 2006, room 526

Speaker:** Yurii Lyubarskii **(NTNU, Trondheim)

"On decay of holomorphic functions"

Wednesday, March 29, 2006, room 526

Speaker: **Irina Markina** (UiB)

Title: "About Heisenberg group" (final talk)

Wednesday, March 22, 2006, room 526

Speaker: **Irina Markina** (UiB)

Title: "About Heisenberg group" (continuation)

Wednesday, March 15, 2006, auditorium Pi

Joint Analysis seminar and Department's colloquium

Speaker: **Björn Gustafsson** (KTH, Stockholm)

Title: "On inverse balayage and potential theoretic skeletons"

Wednesday, March 8, 2006, room 526

Speaker: **Irina Markina** (UiB)

Title: "About Heisenberg group"

Thursday, February 2 and 16, 2006, room 510

Speaker: **Alexander Vasiliev** (UiB)

Title: "Bosonic strings and subordination evolution"

### 2005

Thursday, December 8, 2005, room 526

Speaker: **Arne Stray** (UiB)

Title: "A problem about harmonic functions"

Thursday, October 13, 2005, room 526

Speaker: **Alexander Vasil'ev** (UiB)

Title: "Modeling 2-D flows in Hele-Shaw cells by conformal maps"

Thursday, September 22, 2005, room 526

Speaker: **Giuseppe Coclite** (CMA Oslo and University of Bari, Italy)

Title: "Global Weak Solutions to a Generalized Hyperelastic-Rod Wave Equation"