# Program

## PROGRAM

### 22.03.2013

** (Uniwersytet Warszawski) Piotr Soltan**

*Around Gelfand's theorem*

I will recall various aspects of the theorem of Gelfand providing equivalence between categories of commutative C*-algebras and that of locally compact topological spaces. After analyzing its modern formulation I will pass to an interesting theorem of S.L. Woronowicz which shows how to recover information about a C*-algebra from its, so called, multiplier algebra. Applying this result to commutative algebras yields the fact that a locally compact metrizable space is determined by its Cech-Stone compactification.

### 15.03.2013

**Djalil Chafai (Université Paris-Est Marne-la-Vallée)**

*Around the circular law*

Among basic results of random matrix theory, the circular law is probably the simplest to state and the hardest to prove. This talk will present the circular law and various related models and topics.

### 8.03.2013

**Galina Filipuk (Uniwersytet Warszawski)**

*Introduction to orthogonal polynomials*

Orthogonal polynomials appear in many areas of mathematics and mathematical physics. In this talk I shall explain some of the main properties of orthogonal polynomials. The coefficients of the three-term recurrence relation are of particular interest.

### 1.03.2013

**Grzegorz Sójka (Politechnika Warszawska)**

*Outline of Geometric Tomography*- At the beginning of the talk we will give some information about history of GT. Then we will describe basic notions and typical research techniques by presenting early problems and their solutions (e.g. equichordal point problem, Petty-Busemann problem, etc.).

### 22.02.2013

**Ewelina Seroka (UKSW)**

*Stability of stochastic hybrid systems*

In my talk I would like to present the basic definitions and concepts of deterministic and stochastic hybrid systems. In particular, I will discuss the problem of stability and stabilizability of stochastic hybrid systems. I will present the basic problems that arise in the study of the stability and stabilizability of hybrid systems, and that did not occur in the case of non-hybrid systems. At the end I will formulate sufficient conditions for the stabilizability of nonlinear and bilinear stochastic hybrid systems with Markovian switching.

### 25.01.2013

**Piotr Oprocha (AGH, Kraków)**

*Recurrence in pairs - known results and open problems*

Consider a dynamical system on a compact metric space. Generally speaking, a point x is product recurrent if its returns to any open neighborhood of x can by synchronized with returns of any other recurrent point (in any other dynamical system). In other words, x in pair with any other recurrent point y can return simultaneously to their respective neighborhoods. Points x with the above property (the so-called product recurrent points) have been fully characterized many years ago. If we weaken assumptions on synchronization (e.g. we demand synchronization with recurrent points y from some specified class of dynamical systems), then we can obtain a larger class of admissible points x. In this talk we will present some recent results and open problems on product recurrence and related topics.

### 18.01.2013

**Adam Osękowski (MIM UW)**

*UMD spaces, their properties and applications*

UMD Banach spaces (Unconditional for Martingale Differences) is a class of Banach spaces which arises naturally in the study of Lp boundedness of transforms of vector valued martingales. It turns out that this class forms a convenient environment in harmonic analysis: for example, it can be shown that many classical singular integral operators are Lp-bounded (p strictly between one and infinity) on UMD-valued functions. During the talk, we will study some basic properties of UMD spaces and mention several applications.

### 11.01.2013 - Room 322

**Piotr Zgliczyński (Jagiellonian University)**

*Rigorous computer assisted results in the N-body dynamics*

After the introduction to the N-body problem and the restricted three body problem, we will discuss several computer assisted proofs of dynamics in the N-body problem and the planar restricted three body problem. The main examples discussed will be: - the existence of choreographies in the N-body problem with N > 2, the choreography is a solution where all bodies move on the same path - resonance transitions for Oterma comet in the Sun-Jupiter system Other topics that will be briefly mentioned: - invariant tori, KAM theory, Arnold diffusion I intend to keep the talk on rather elementary level.

### 14.12.2012

**Armen Edigarian (Jagiellonian University)**

*Some special domains in complex analysis*

The main idea of the talk is to present two domains intensely analysed within the last few years: the symmetrised bidisc and the tetrablock. In 1981 Lempert proved that in convex domains invariant metrics are equal. For almost 25 years the role of convexity was unclear. Only the recent analysis of the symmetrised bidisc, a very simple domain, opened new areas of study.

### 7.12.2012

**Mikołaj Bojańczyk (MIM UW)**

*Computing with Atoms*

The talk is about an approach to algorithms that process infinite systems. The approach is to use a different set theory, namely Fraenkel-Mostowski sets theory (also known under the following names: nominal sets, sets with ur-elements, sets with atoms, permutation models). In Fraenkel-Mostowski, there are more sets than in usual set theory, but what is most important to us, there are more finite sets. Even though the notion of finiteness is more relaxed, it is still possible to do a lot of computer science, such as programming, automata or logic.

### 31.11.2012

**Balazs Barany (IM PAN)**

*How to calculate the dimension of some fractal sets?*

In this talk I would like to present a family of fractal sets, the so-called self-similar sets. These sets are generated by a set of contracting similarity functions. The set of the functions is called Iterated Function System (IFS). Several famous sets are self-similar, for example the Sierpinski gasket, the Koch curve and the Cantor set. One of the important properties of these sets is the dimension.The calculation is relatively easy when we have some separation between the images. Otherwise, if we cannot guarantee any separation we have only "typical" results about the dimension. The goal of the talk is to show some methods how to calculate the dimension of self-similar sets in both cases.

### 23.11.2012

**Maciej Dołęga (Uniwersytet Wrocławski)**

*Representation theory, symmetric functions and bipartite maps*

I would like to explain how characters of symmetric groups can be generalized using theory of symmetric functions. Doing that, I define a one-parameter deformation of characters of symmetric groups called Jack characters. Later on, I am going to present a conjectural formula expressing Jack characters using combinatorics of some nice, geometric objects, namely bipartite maps. Finaly I am going to present consequences of this conjectural formula as well as explain some heuristic that leads us to believe our conjecture is true.

### 16.11.2012

**Andrzej Nagórko (IM PAN)**

*Infinite-dimensional topology*

Infinite-dimensional topology is a branch of geometric topology devoted to the study of manifolds modelled on the Hilbert cube and the Hilbert space. I'll survey classical theory and talk about recent work on its finite-dimensional counterparts.

### 9.11.2012

**Javier de Lucas (Uniwersytet Kardynała Stefana Wyszyńskiego)**

*Geometric structures in Lie systems*

A Lie system is a system of first-order differential equations admitting a superposition rule, i.e. a map that expresses its general solution in terms of a generic finite family of particular solutions and some constants. In this talk, we will mainly present the most basic features of Lie systems. We will discuss the Lie-Scheffers theorem, which characterises systems possessing a superposition rule, and will show how it leads to the problem of classifying finite-dimensional real Lie algebras of vector fields on manifolds. Next, we will describe the Lie group and distributional approaches to Lie systems. Finally, we will briefly analyse some recent applications of Poisson geometry and co-algebras in Lie systems. Many new examples of Lie systems occuring in physics and mathematics will be detailed and possible generalisations, e.g. to non-commutative geometry, will be commented.

### 26.10.2012 - Room 322

**Andrzej J. Maciejewski (Uniwersytet Zielonogórski)**

*N-body problem in mathematics and astronomy: dynamics of old and new discovered systems*- Wyklad poswiecony jest omowieniu podstawowych wlasnosci zagadnienia N cial oraz problemow matematycznych z nim zwiazanych. Rozpatrzymy klasyczne grawitacyjne zagadnienie N punktow materialnych oraz rozne jego warianty (ograniczone zagadnienia oraz zagadnienie Hilla). Przedstawie problem konfiguracji centralnych oraz problem osobliwosci niezderzeniowych. Jednym z waznych zastosowan zagadnienia N cial jest badanie dynamiki i wyznaczanie parametrow planet oraz ich orbit w pozaslonecznych ukladach planetarnych. Przedstawie problemy zwiazane z tego typu zastosowaniami. Istotna role w zastosowaniach odgrywaja rowniez warianty zagadnienia N cial, w ktorych jedno lub kilka cial sa obiektami rozciaglymi (np. brylami sztywnymi). Zagadnienia dwoch bryl sztywnych stosuje sie na przyklad do badania dynamiki podwojnych asteroidow. Pokaze, jak jakosciowo zmienia sie skala trudnosci, gdy przechodzimy od zagadnien punktowych N cial do zagadnien rozciaglych N cial.

### 19.10.2012

**Piotr Przytycki (IM PAN)**

*Dismantlability*

I will explain what it means for a finite graph to be "dismantlable" and sketch a proof of Polat's theorem that such a graph has a clique fixed under all of its automorphisms. I will present applications to various infinite graphs appearing in geometric topology.

### 12.10.2012

**Jan Rudnik (IM PAN)**

*Noncommutative bundles over the triple-Toeplitz deformation of the real projective plane*

I will show how one can use well known constructions from classical topology to obtain quantum analogs of the real projective plane. Then we will construct Z2 principal bundle over it and prove a non-triviality result about the tautological line bundle.