MPD - Geometry and Topology in Physical Models

Detailed information about past talks (2011/2012)

Date	Information
04.10.2011	Title: "Spectral Triples and Physics" Speaker: Michal Eckstein Abstract: In my talk I will present the main ingredients of noncommutative differential geometry la Connes and some of its physical applications. In particular, I will address the problem of quantum anomalies in noncommutative spaces.
11.10.2011	Title: "Bergman completeness is not a quasi-conformal invariant" Speaker: Włodzimierz Zwonek (after Xu Wang)
18.10.2011	Title: "Scalar perturbation of spectral action" Speaker: Artur Zając Abstract: Spectral action is the noncommutative analogue of classical action functional. However, in noncommutative regime there are possible new types of action perturbation, which occur trivial in commutative case. The simplest case of such perturbation is the scalar one. I will show our recent results concerning scalar perturbation, obtained by Andrzej Sitarz and me.
08.11.2011	Title: "Moser-Trudinger inequality and some applications" Speaker: Nguyen Ngoc Cuong Abstract: We will discuss the original inequalitty due to Neil Trudinger in 1967 and a sharp form of this by J.Moser. It was originally motivated by critical case of embedding Sobolev theorem and from that time, several authors had found many applications of this inequality, in PDE and especiallly in complex geometry (existence of Kahler-Einstein metric on Kahler manifold).
15.11.2011	Title: "Computational (persistent) homology with applications" Speaker: Hubert Wagner Abstract: In this talk I would like to present some concepts from the field of computational topology. I'll focus on the so-called persistent homology, which is a modern extension of the well-known homology theory. This topic will be presented from the perspective of algorithms and applications.
22.11.2011	Title: "Homotopy representable functors for differential topologist" Speaker: Andrzej Czarnecki Abstract: We will see how the h-principle allows classification of foliations and microbundles via homotopy theory. This clasical framework is one of the central points of current development of foliations theory.
06.12.2011	Title: "Number-theoretical phenomena in dynamics" Speaker: Piotr Kamieński Abstract: Behavior of dynamical systems is very often influenced by arithmetic properties of some numbers related to them. We will discuss theorems linking two seemingly distant areas of mathematics - number theory and dynamics.
13.12.2011	Title: "Heat flow for harmonic maps" Speaker: Paweł Biernat Abstract: I will describe the idea of a heat flow as a tool for finding critical points of a functional. The harmonic map heat flow is a flow for a Dirichlet energy of maps between two compact Riemannian manifolds. I will focus on a heat flow for maps between two spheres of the same dimension d>2.
20.12.2011	Title: "Critical phenomena" Speaker: Joanna Jałmużna Abstract: In my talk I will give an introduction to the critical phenomena for nonlinear differential equations and then I will discuss it in the context of Einstein's equations and gravitational collapse.
03.01.2012	Title: "Uniqueness and existence theorem of Dirichlet problem for the complex Monge-Ampere operator" Speaker: Frunza Ataev Abstract: The main goal of my talk is to present uniqueness and existence theorem of Dirichlet problem for the complex Monge-Ampere operator which firstly was proved by E. Bedford anad B. A. Taylor in 1982. I will also present some other recent results. We know that, the complex Monge-Ampere operator is analogue of the Laplace operator. Monge-Ampere equations frequently arise in differential geometry. They were discovered by Gaspard Monge in 1784 and later by Andre-Marie Ampere in 1820.
10.01.2012	Title: "The dimension spectrum of quantum spheres" Speaker: Michał Eckstein Abstract: In the theory of noncommutative spaces (in the sense of spectral triples) the notion of dimension spectrum plays an important role. For a classical manifold it reduces to the standard definition of the dimension, but in general it may be an arbitrary discrete subset of the complex plane. In my talk I will present the preliminary results concerning the dimension spectrum of the standard quantum sphere. Moreover, I will show how the introduction of an auxiliary twist affects the dimension spectrum.
24.01.2012	Title: "Extension of separately holomorphic functions on generalized (N,k)-crosses" Speaker: Arkadiusz Lewandowski Abstract: The famous Hartogs theorem states that any separately holomorphic function in several complex variables is necessarily holomorphic as a function of all variables. In our talk we sketch the theory of extension of separately holomorphic functions from this deep result of Friedrich Hartogs to the most recent results concerning separately holomorphic functions on the so-called generalized (N,k)-crosses (also with singularities).
28.02.2012	Title: "The Suita Conjecture" Speaker: Zbigniew Błocki Abstract: We will present a conjecture formulated by N. Suita in 1972 which postulates an optimal estimate between the logarithmic capacity and the Bergman kernel for domains on the plane. We will discuss some known results in this direction, the relation of this conjecture to the Ohsawa-Takegoshi extension theorem and Hoermander's L^2-estimate for the \bar\partial-equation. We will also present a new result which reduces this conjecture to an ODE problem as well as some partial numerical results in this direction.
06.03.2012	Title: "Some remarks on Costara's paper 'On the spectral Nevanlinna-Pick problem'" Speaker: Maria Trybuła Abstract: I reformulate theorems satated by Costara in terms of polar derivative.
13.03.2012	Title: "Asymptotic of blow-up for harmonic map flow between Spheres of dimensions d>2" Speaker: Paweł Biernat Abstract: I will show how to construct corotational solutions to the harmonic map flow between d-Spheres which are generic and blow-up at finite time.
20.03.2012	Title: "Scalar parabolic dynamics and braids" Speaker: Aleksander Czechowski Abstract: We will study spatial discretization of scalar reaction diffusion equation. It is well known that the number of intersections between solution curves of such system does not increase in time. This feature can be exploited to construct new solutions. We will present suitable methods in the geometric language of braids.
27.03.2012	Title: "Introduction to the basics of differential Galois theory" Speaker: Piotr Kamieński Abstract: I will give a brief overview of the Galois theory for linear differential equations, discuss some important results and mention possible applications, in particular to Hamiltonian systems.
17.04.2012	Title: "J-Flow in complex geometry" Speaker: Nguyen Ngoc Cuong Abstract: At the same time S. Donaldson and X.X. Chen (about 1999-2000) have discovered a flow which could be considered as a moment map (Donaldson) or as a gradient flow of a functional (Chen). The motivation is to understand the space of Kahler potential of a given cohomology class on Kahler manifold. In this talk I will try to explain this flow (name J-flow) in point of view given by X.X. Chen and discuss the results in studying this flow recently by other authors (e.g. B. Weinkowe, J. Song) and its role in Kahler geometry if it is possible.
17.04.2012	Title: "Heat kernel expansion" Speaker: Artur Zając Abstract: I will explain, how to extract from the Dirac operator on a spin manifold some geometrical data using heat kernel expansion. I will also explain the connection of heat kernel coefficients with zeta function of an operator. Finally I will show the use of heat kernel expansion in noncommutative geometry. I will also mension briefly my latest calculations of of heat kernel coefficients on spherical space forms.
08.05.2012	Title: "On Spatial Contagion and mGARCH models" Speaker: Marcin Pitera Abstract: My talk will be based on a preprint written jointly with prof. P. Jaworski. I will propose a method for defining and measuring the spatial contagion between two financial markets. Next I investigate which from the large family of multivariate GARCH models is the best tool for modeling spatial contagion.
22.05.2012	Title: "A computer-assisted proof to detect chaos" Speaker: Frank Weilandt Abstract:I will present a method developed my Prof. Mrozek (my advisor) and Prof. Srzednicki that can detect chaos in certain non-autonomous, time-periodic differential equations (simple intuition: vector fields in the complex plane changing in time). It was inspired by the machinery developed by Ważewski and Conley for topological dynamics. A purely topological theorem is adapted to the needs of rigorous numerics.
29.05.2012	Title: "Invariant metrics and distances. Lempert's theorem." Speaker: Arkadiusz Lewandowski Abstract: We shall discuss invariant metrics and distances in complex analysis. The Caratheodory and the Kobayashi pseudodistances as well as the Caratheodory-Reiffen and the Kobayashi-Royden pseudometrics will be of our particular interest. We present a very deep and beautiful result due to L. Lempert which says that the Kobayashi and Caratheodory pseudodistances coincide on some class of domains.
05.06.2012	Title: "The Evans-Krylov theory for fully nonlinear equations" Speaker: Gu Dongwei Abstract: I'd like to talk about some general theory on PDEs. I will introduce some background on this theory and then explain its importance in solving nonlinear equations.I will also show some examples so that we can see its usefulness.
12.06.2012	Title: "Measures of risk - general introduction" Speaker: Marcin Pitera Abstract: I'd like to talk about measurement of risk. We will introduce general concept of risk measures, showing the evolution of that idea. Starting from variance, sem-ivariance, going through the concept of Value at Risk, spectral and coherent risk measures, we will end with dynamic framework.
09.10.2012	Title: "The super-potential of closed positive currents and the complex Monge-Ampere equation" Speaker: Nguyen Ngoc Cuong Abstract: The notion of the super-potential for closed positive currents with arbitrary degree was introduced and developed by T-C. Dinh and N. Sibony. It has numerous applications in the intersection theory and complex dynamics in higher dimension. On the other hand, the complex Monge-Ampere equation on a compact Kahler manifold is the classical subject but it is still a very active research domain. A question related to weak solutions (Holder continuous) is to find a good characterization of the measure on the right hand side such that there exists a Holder continuous solution (i.e potential in a sense). Using the super-potential notion, V-A. Nguyen and T-C. Dinh have recently shown that "The complex Monge-Ampere equation on a Kahler manifold admits a Holder continuous quasi-plurisubharmonic function if and only if its right hand side is a positive measure (closed positive current of highest bidegree) with Holder continuous super-potential". In this talk I will discuss some results about the Holder continuous solution to the complex Monge-Ampere equation and the above result.
16.10.2012	Title: "Forms, Homotopy, Logarithms" Speaker: Andrzej Czarnecki Abstract: I will say few words about advantages of using homotopy theory instead of analysis or point set topology when dealing with some analytical problems. A motivational example will be a theorem characterising trivial first cohomology group on manifolds.
23.10.2012	Title: "Real Hessian equation" Speaker: Frunza Ataev Abstract:I would like to talk about real k-Hessian equation. I will present Dirichlet problem for real k-Hessian equation. I will also show some geometric applications of k-Hessian equations, which have been well developed by J.Spruck and others. Hessian equations have been extensively studied in the recent years. A great impetus had been given because of the paper written by Caffarelli, Nirenberg and Spruck. Also various papers written by Trudinger and Wang had a major impact in the increase of interest on that field.
30.10.2012	Title: "Self-maps of spectral unit ball" Speaker: Maria Trybuła Abstract: I will show some theorems about group automorphisms of spectral unit ball.
06.11.2012	Title: "Siaciak-Zahariuta extremal function, analytic discs and polynomial hull" Speaker: Ibrahim Djire Abstract: In this talk I give two disc formulas for the Siciak-Zahariuta extremal function of an arbitray open subset of complex affne space and use these formulas to characterize the polynomial hull of an arbitry compact subset of complex affne space in terms of analytic discs.
13.11.2012	Title: "The Spectral Action Principle " Speaker: Michał Eckstein Abstract: The spectral action is a way to construct physical models of elementary interactions in the general framework of noncommutative geometry. In my talk I will present the rudiments ilustated by examples as well as my own results. The talk is aimed at non-specialists.
20.11.2012	Title: "Dimension spectrum" Speaker: Artur Zając Abstract: -
27.11.2012	Title: "Performance Measures and Acceptability Indices" Speaker: Marcin Pitera Abstract: We will characterize special class of performance measures known as Acceptability Indices, which were introduced by A. Cherny and D. Madan. We will also explain why they are strictly connected with coherent risk measures.
04.12.2012	Title: "Some equivalent formulations of PDE problems" Speaker: Gu Dongwei Abstract: On any compact Kahler manifold, the space of Kahler metrics is a Riemannian manifold. It's interesting to consider the space of volume forms on any compact Riemannian manifold. It has a geometry structure as well. The geodesic equation and its perturbed equations are equivalent to some Laplacian equations and have a relationship with other problems,such as Nahm's equation.
11.12.2012	Title: "Numerical calculation of Conley indices and Morse decompositions for dynamical systems" Speaker: Frank Weilandt Abstract: Considering solution curves of ordinary differential equations leads to the notion of dynamical systems. For such a system, an invariant set is a subset of the phase space in which solutions stay for all time. The type of an invariant subset (e.g. a fixed point) can be classified using the Conley index, defined by homology groups or the homotopy type of a certain space. It gives coarse information about the behavior of solutions in a neighborhood of this set. It cannot, however, tell us completely what happens inside the invariant set. A more detailed description of the behavior of solutions inside an invariant set is given by Morse decompositions. An invariant set can contain strict subsets that are themselves invariant. The Morse decomposition gives a description of relations between several invariant subsets. My talk will introduce these two descriptors of dynamical systems, and how we can numerically find Morse decompositions and the Conley indices of their invariant sets. The algorithms are using the CAPD library from UJ and a database project maintained by Rutgers University.
18.12.2012	Title: "Gravitational collapse in AdS spacetimes" Speaker: Joanna Jałmużna Abstract: In my talk I will try to explain why, from physical point of view, this subject is important. I will start with introduction to General Relativity, gravitational collapse and AdS spacetime. Finally, I will present a model problem I am working on and I will show why numerical simulations can help us understand the nature of the problem.
08.01.2013	Title: "Some property of the Bergman Kernel" Speaker: Maria Trybuła Abstract: I will present the well known property of the Berman Kernel in a new way.
05.03.2013	Title: "The viscosity solutions to complex Monge-Ampere equations and complex Hessian equations" Speaker: Nguyen Ngoc Cuong Abstract: Since the viscosity solution concept was introduced by L. Lion and M. Crandall in the early 1980 as the generalization of the classical solution concept of partial differential equations (PDEs),it has been become the central concept in the study of elliptic PDEs. It also has been found that the viscosity solution is the natural solution concept to use in many applycations of PDEs. This method recently has been applied to the study of complex Monge-Ampere equations and of the complex Hessian equations. In this talk I would like to discuss the notion of the viscosity solution in general through examples, and then I will talk about recent results on the viscosity solution to the Dirichlet problem for complex Monge-Ampere equations and for complex Hessian equations in a domain in C^n.
12.03.2013	Title: "Geometry on singular spaces" Speaker: Andrzej Czarnecki Abstract: We will present several approaches how to construct geometric structures on orbifolds, the simplest examples of singular spaces. The theme will be to use transverse geometry on an appropriate foliated space.
19.03.2013	Title: "Bourgain-Milman inequality and Mahler conjecture for convex symmetric bodies" Speaker: Zbigniew Błocki Abstract: We will present several approaches how to construct geometric structures on orbifolds, the simplest examples of singular spaces. The theme will be to use transverse geometry on an appropriate foliated space.
26.03.2013	Title: "Effective diophantine approximation" Speaker: Piotr Kamieński Abstract: I will explain why dynamicists need effective diophantine approximation - i.e. effective computation of the diophantine exponent and constant for a given irrational number - and describe how to obtain these quantities for quadratic irrationals with known continued fraction expansion.
09.04.2013	Title: "Henstock-Kurzweil integral" Speaker: Arkadiusz Lewandowski Abstract: We present the Henstock-Kurzweil integral, which is a simultaneous, nontrivial generalization of Riemann integral, Lebesgue integral, as well as improper Riemann integral. We discuss some of its properties and possibilities of further generalization, as well.
16.04.2013	Title: "Gravitation theory in 3-dimensional space" Speaker: Joanna Jałmużna Abstract: Gravitation theory in 3 dimensions is important from the point of view of quantum theory of gravity. In my talk I will shortly remind you main concepts of Einstein's theory of gravity. Later part will be based mainly on prof. Staruszkiewicz's paper about gravitation theory in three-dimensional spacetime (with two space-like and one time-like dimension). I will describe a general solution of the problem of motion of point particles with finite mass.
23.04.2013	Title: "Moser's theorem about volume forms on a manifold" Speaker: Gu Dongwei Abstract: If a smooth manifold has two different volume forms, one can ask whether the first is equivalent to the second via a pullback by some diffeomorphism. A necessary condition is that the forms must have the same total volume. MoserÇs theorem states that this is also a sufficient condition: any volume form can be pulled back to another with the same total volume by some diffeomorphism. The proof of this theorem is interesting and basic.
30.04.2013	Title: "Graphs that are not complete pluripolar" Speaker: Ibrahim Djire Abstract: Let A be a subdomains of B in the complex plane. Under very mild conditions on B we show that there exists holomorphic functions f, defined on A with the property that f is nowhere extendible across the boundary of A, while the grath of f over A is not complete pluripolar.
14.05.2013	Title: "Atiyah-Singer index theorem revisited" Speaker: Paul Baum (Pennsylvania State University) Abstract: This is an expository talk about the Atiyah-Singer index theorem. The talk will consist of four points: Some low dimensional examples of the theorem will be considered. Definition and construction of the Dirac operator. A special case of the theorem will be proved, with the proof based on Bott periodicity. The proof will be outlined that the special case implies the full theorem. This is part of an index theory project with Erik van Erp.
21.05.2013	Title: "Group gradings on Lie algebras" Speaker: Mikhail V. Kotchetov (Memorial University of Newfoundland) Abstract: The Cartan decomposition of a semisimple Lie algebra with respect to a Cartan subalgebra can be regarded as a grading by a free abelian group. Gradings on Lie algebras by various abelian groups arise in the theory of symmetric spaces, Kac-Moody algebras, and color Lie superalgebras. In the 1960s, V. Kac classified all gradings by cyclic groups on finite-dimensional simple Lie algebras over complex numbers. We will discuss recent progress in the classification of gradings on classical simple Lie algebras by arbitrary groups.
28.05.2013	Title: "The Mellin transform" Speaker: Artur Zając Abstract: I shall define an integral transform called Mellin transform. I will discuss its general properties and relations with other well known integral transforms (Fourier, Laplace). Then I will say a few words about its applications in calculating operator spectra and traces.
11.06.2013	Title: "Efficient computation of persistent homology for image data using discrete Morse theory" Speaker: Hubert Wagner Abstract: We describe an efficient algorithm for computing persistent homology of 3D image data. Using Morse-Smale complex preprocessing together with an implicit representation of cubical complexes, memory usage is significantly reduced. This is an important improvement, as practical applications involve images containing at least 10^9 voxels. In such situations explicitly storing the boundary matrix of the input complex is prohibitive. We show that our method overcomes this difficulty, while being parallelizable and efficient in practical situations.
11.06.2013	Title: "An algorithm for computing the Conley index of a Poincare map" Speaker: Frank Weilandt Abstract: Non-autonomous periodic ODE's generate a flow (the solution curves considered together), and they give rise to Poincare maps. The homological Conley index of a Poincare map provides essential information on the qualitative behavior of the flow. In particular, it can be applied to prove the existence of periodic orbits or chaotic dynamics. In order to determine the index, one usually numerically follows trajectories of the flow. That causes technical difficulties due to exponentially growing errors of calculations. Our new approach requires only the numerical integration of the flow for a small time. I will present Conley index theory and our approach to calculate the Conley index for certain Poincare maps.
15.10.2013	Title: "Lipschitz stability for compactly supported scalar potentials in an infinite quantum waveguide" Speaker: Phan Quang Sang Abstract: We consider the inverse problem of determining the time independent and a priori known outside some fixed compact set scalar potential of the dynamic Schrdinger equation in an infinite cylindrical domain from one boundary Neumann observation of the solution. This is obtained with the help of a new gobal Carleman estimate designed specipically for the Schrdinger operator acting in an unbounded cylindrical domain, Using a Carleman estimate for the Schrdinger equation in a bounded domain we obtain Lipschitz stability in the determination of the scalar potential from both bounded boundary and arbitrarily small (with respect to the infinite direction of the waveguide) internal measurements of the solution.
22.10.2013	Title: "Small denominators and the Khintchine-Levy constant" Speaker: Piotr Kamieński Abstract: The theory of stability of quasiperiodic motions under small perturbations of is an important part of Hamiltonian dynamics. I will approach the problem in a statistical manner and using number-theoretical tools, which will result in KAM-like theorems.
29.10.2013	Title: "Topological methods in slow-fast systems" Speaker: Aleksander Czechowski Abstract: Many of the real-world phenomena are modeled by systems of differential equations in which velocity scales between variables vastly differ. Simulation of these so-called slow-fast systems is hard to perform using standard numerical methods. We will show how one can employ topological methods together with rigorous numerics to prove results about the existence of periodic motions. Our model example will be the FitzHugh-Nagumo system of the electric impulse propagation in an axon.
12.11.2013	Title: "On the heat kernel" Speaker: Michal Eckstein Abstract: Given a positive, possibly unbounded, operator A on a separable Hilbert space one can define the associated heat operator e^{-t A}, which, under some mild conditions on A, is trace-class for any t>0. It turns out that a close inspection of its kernel, i.e. the function t --> Tr e^{-t A}, reveals a lot of information of geometrical nature. During the seminar I will introduce the necessary rudiments and present some new results based on a join project with Artur Zajac.
19.11.2013	Title: "On the heat kernel 2" Speaker: Artur Zajac Abstract: It will be a continuation of Michal Eckstein's talk from the previous week. I will present our main result on the existence of heat kernel expansion, with a set of sufficient conditions. These conditions will be explained, and a sketch of the proof will be provided. Then I will present a handful of examples, most of them geometrically motivated.
26.11.2013	Title: "Analytic Discs, Global Extremal Functions and Projective Hulls in Projective Space." Speaker: Ibrahim Djire Abstract: Using a recent result of Larusson and Poletsky regarding plurisuharmonic subextensions we prove a disc formula for the quasiplurisubharmonic global extremal function for domain in $\P^n$. As a corollary we get a characterization of the projective hull for connected compact sets in $\P^n$ by the existence of analytic discs.
03.12.2013	Title: "Three-dimensional gravity and AdS_3 instability" Speaker: Joanna Jałmużna Abstract: I will describe main properties of Einstein's equations in three dimensions and present the problem of stability of Anty-de Sitter (AdS) spacetime. On the example of three-dimensional asymptotically AdS spacetime I will present how the mixture of numerical and analytical approaches can help and lead to conjecture formulation.
14.01.2014	Title: "Weak solutions to the complex Hessian equations" Speaker: Nguyen Ngoc Cuong Abstract: The complex Hessian equation is a natural generalisation of the complex Monge-Ampere equation where weak solutions have found many geometrical applications. This is a strong motivation to construct the weak solution theory for the complex Hessian equation. First I will some background on the development of the complex Hessian equation in a domain in C^n. Then, I will explain how a priori estimates for the Monge-Ampere equation derive a priori estimates for the complex Hessian equation. Finally, I will give the summary of results of my thesis which all focus on bounded, continous and Holder continous solutions.
21.01.2014	Title: "Engineering computational topology algorithms" Speaker: Hubert Wagner Abstract: In this talk I focus on practical performance of computational topology algorithms, especially the classical, boundary matrix reduction algorithm for computing persistent homology. This algorithm works for arbitrary filtered chain complexes. I show that careful optimizations increase the practical efficiency by several orders of magnitude. In particular, a new strategy for accumulating columns will be presented. It is based on a data-structure we call a bit-tree, which is tailored especially to this application.
28.01.2014	Title: "Dynamic Limit Growth Indices in Discrete Time" Speaker: Marcin Pitera Abstract: The talk will be based on a joint work with prof. I. Cialenco and T.R. Bielecki (IIT). We will discuss a new class of mappings, called Dynamic Limit Growth Indices, that are designed to measure the long-run performance of a financial portfolio in discrete time setup. We will study various important properties for this new class of measures, show some examples and discuss time consistency property.
25.02.2014	Title: "Time discretization strategies for finding Morse decompositions of flows" Speaker: Frank Weilandt Abstract: Morse decompositions are useful for describing the long-term behavior of the solutions of ordinary differential equations. They can often be found numerically in a rigorous sense for a computer-assisted proof. There are well-established methods for finding Morse decompositions for discrete dynamical systems (iterations of continuous maps). The solutions of an ODE give us such a discrete system by "jumping" along each solution curve. But how far to jump involves a choice. I will present examples where we want some flexibility for this choice and the theory to achieve that.
04.03.2014	Title: "About openness conjecture" Speaker: Gu Dongwei Abstract: It was solved recently by several people. I will explain some basic concept related to this conjecture and mainly deal with the 1-dimension case. In general case, it can be proved by Ohsawa-Takegoshi extension theorem.
11.03.2014	Title: "Spin and Spin^C structures on flat manifolds" Speaker: Andrzej Szczepański (University of Gdańsk) Abstract: Flat manifolds are compact Riemannian manifolds with sectional curvature equal to zero. Any such manifold is the quotient space R^n/G, where G is a torsion free, discrete and cocompact subgroup of the isometry group of the Euclidean space R^n. We shall define and descibe the Spin and Spin^C structures on such objects.
18.03.2014	Title: "Invariant structures on Lie groups with some applications to symplectic foliations" Speaker: Andrzej Czarnecki Abstract: A philosophical question: does a group of isomorphisms of a structure (metric, complex...) carry a natural structure of the same type? It is always true for metric, it is never true for complex structure. It is sometimes true for symplectic structure. I will discuss this topic and then apply the invariant symplectic structure to calculation of basic version of S.T. Yau's symplectic cohomologies.
25.03.2014	Title: "Construction of a formal blow-up solution to harmonic map heat flow" Speaker: Pawel Biernat Abstract: Heat flow for harmonic maps is known to produce solutions for which the first derivative blows-up in finite time. These singular solutions arise for a large class of initial data and present a major obstacle in solving this heat flow equation for arbitrarily large times. A typical solution with blow-up in its derivative evolves by rescaling (i.e. it shrinks around a fixed point). The speed of this process is called a blow-up rate. I will demonstrate how to construct a (formal) solution for which the blow-up rate is not the one we would expect from a parabolic equation. My construction is based on the matched asymptotic technique coming from ordinary differential equations.
01.04.2014	Title: "Quantum information, entanglement and geometry" Speaker: Ana Kontrec Abstract: Quantum information theory investigates the physics of (quantum) information. One of the central notions and a fundamental resource is quantum entanglement - the phenomenon of non-classical correlations. In this talk I will introduce some underlying concepts from quantum theory and try to give a geometrical perspective on problems in entanglement theory and quantum information processing.
08.04.2014	Title: "Rigorous numerics for partial differential equations" Speaker: Aleksander Czechowski Abstract: I will talk about how to do rigorous computations and prove existence of invariant objects, such as stationary or periodic solutions, in nonlinear dissipative PDEs. In particular, I will discuss the framework of self-consistent bounds developed by Mischaikow and Zgliczynski, which allows to reduce the system to a finite-dimensional differential inclusion.
15.04.2014	Title: "Classical monodromy" Speaker: Quang Sang Phan Abstract: There exists locally action-angle coordinates for a completely integrable systems, shown by Liouville- Mineur-Arnold theorem. However, we have maybe no global existence of these coordinates, for example the spherical pendulum or the bottle of champagne . A obstruction of this is a geometrical invariant, called monodomy, given by Duistermaat. Moreovere, the Liouville torus fibration has an intergal structure, in general non-trivial.
29.04.2014	Title: "Littlewood's Subordination Principle" Speaker: Ibrahim Djire Abstract: My goal is to present the subordination principle and some of its consequences (for more details see P. L. Duren's book : Univalent functions, chap. 6). Assuming that, the superordinate function is proper I will prove that the subordination principle can be generalized to the situation where the functions are not globally subordinate.
06.05.2014	Title: "On the behaviour of holomorphically invariant distances on special classes of domains" Speaker: Maria Trybula Abstract: I will present some new recent results by Nikolov, Pascal and Trybula concerning holomorphically invariant distances.
03.06.2014	Title: "Probabilistic number theory methods in Hamiltonian dynamics" Speaker: Piotr Kamienski Abstract: In Hamiltonian systems quasiperiodic motions confined to invariant tori are important objects of study. When an integrable Hamiltonian system is perturbed some of the tori, which originally filled up almost the whole phase space, are destroyed and the remains form a Cantor-like set. In the talk I will show a new method to determine the measure of this set, which uses metrical theory of continued fractions and the theory of large deviations of random variables.
10.06.2014	Title: "Rearrangement and applications in PDEs" Speaker: Zbigniew Blocki Abstract: For a nonnegative measurable function its rearrangement (or Schwarz symmetrization) is the unique radially symmetric function which is non-increasing (in radius) and whose super-level sets have the same measure as the initial function. It turns out to be a very useful notion, in particular for some elliptic PDEs.

Geometry and Topology in Physical Models

International PhD Programme in Mathematics, Mathematical Physics and Computer Science