Analysis Group Seminar

RECENT SEMINARS

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 S¹ and on weighted Bergman spaces over a domain Ω ⊂ Cⁿ, 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 Cⁿ, 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 Mrolling 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.

ARCHIVE

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 R³"

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 R³, 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 L₁ (R) and L₂(R) in terms of zero sets of Fourier transform.  He conjectured that similar characterization  should be true for L_p (R) , 1<p<2. I will discuss this conjecture. Joint work with Nir Lev.

 

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 Alsaker (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"