### YOUNG RESEARCHERS COLLOQUIUM

 24.05.2013 Tomasz Rogala Construction of discrete time shadow price 17.05.2013 Ada Boralevi Introduction to quiver representations and Gabriel's theorem 10.05.2013 Elisa Postinghel Polynomial interpolation and secant varieties 19.04.2013 Mindaugas Bloznelis Inhomogeneous random intersection graph and process 12.04.2013 Piotr Nayar Discrete cube: combinatorics meets analysis 5.04.2013 Mariusz Mirek Pointwise ergodic theorems along thin subset of primes 22.03.2013 Piotr Sołtan Around Gelfand's theorem 15.03.2013 Djalil Chafai Around the circular law 8.03.2013 Galina Filipuk Introduction to orthogonal polynomials 1.03.2013 Grzegorz Sójka Outline of Geometric Tomography 22.02.2013 Ewelina Seroka Stability of stochastic hybrid systems 25.01.2013 Piotr Oprocha Recurrence in pairs - known results and open problems 18.01.2013 Adam Osękowski UMD spaces, their properties and applications 11.01.2013 Piotr Zgliczyński Rigorous computer assisted results in the N-body dynamics 14.12.2012 Armen Edigarian Some special domains in complex analysis 7.12.2012 Mikołaj Bojańczyk Computing with Atoms 30.11.2012 Balázs Bárány How to calculate the dimension of some fractal sets? 23.11.2012 Maciej Dołęga Representation theory, symmetric functions and bipartite maps 16.11.2012 Andrzej Nagórko Infinite-dimensional topology 9.11.2012 Javier de Lucas Geometric structures in Lie systems 26.10.2012 Andrzej Maciejewski Zagadnienie N ciał i dynamika nowo odkrywanych układów planetarnych/wielogwiezdnych 19.10.2012 Piotr Przytycki Dismantlability 12.10.2012 Jan Rudnik Noncommutative bundles over the triple-Toeplitz deformation of the real projective plane

Tomasz Rogala

I will consider the problem of maximizing discounted expected utility from consumption over finite time horizon in the market with safe bank account a a risky asset (stock) on which an investor pays transaction costs. The problem will be solved using the method of Bellman equations. The main aim of the talk will be to show the existence of so called "shadow price", i.e. a price on the parallel market under which the optimal strategies in both markets are the same. In words, I will show that markets with transaction costs can be studied by the methods we use for studying markets without transaction costs.

Quivers are very simple mathematical objects: finite directed graphs. A representation of a quiver assigns a vector space to each vertex, and a linear map to each arrow. Quiver representations were originally introduced to treat problems of linear algebra, but they soon turned out to play an important role in representation theory of finite dimensional algebras, as well as in less expected domains of mathematics, including Kac-Moody Lie algebras, quantum groups, Coxeter groups, and geometric invariant theory.

In this talk I will introduce quiver representations and the related problem of classification, and give an overview of the celebrated Gabriel's theorem, that gives a very interesting link between quivers and root systems.

Elisa Postinghel

The classical polynomial interpolation theory of functions arises in numerical analysis and statistics and is based on the fact that a single-variable polynomial of degree d is uniquely determined by its values at d+1 distinct points.

The first generalization is the so-called Hermite interpolation that consists of determining the dimension of the vector space of homogeneous or multi-homogeneous polynomials vanishing together with their partial derivatives at a finite set of points.

In this talk I will introduce all the above and relate it with the study of the varieties of secant lines, planes, etc., to a given variety.

Mindaugas Bloznelis

Introduction to the inhomogeneous random intersection graph and process. Degree distribution, clustering and assortativity coefficients of the random intersection graph process.

Piotr Nayar

I will discuss several isoperimetric-type inequalities on the discrete cube {-1,1}n. It turns out that there is a deep relation between combinatorial properties of subsets of the cube and spectral properties of boolean functions. If time permits, I will prove that every symmetric monotone graph property has a sharp threshold behaviour. Our main tools will be Walsh-Fourier expansion and hypercontractivity property of a certain noise-type operator.

Mariusz Mirek

Firstly, I am going to talk about the pointwise ergodic theorems for some subsets of the prime numbers which have relative density 0. Our principal example will be the set of Piatetski-Shapiro primes. Secondly, I will show some connections with additive combinatorics and number theory.

Piotr Sołtan

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 Čech-Stone compactification.

Djalil Chafai

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.

Galina Filipuk

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.

Grzegorz Sójka

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.).

Ewelina Seroka

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.

Piotr Oprocha

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 be 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.

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 (1<p<∞) on UMD-valued functions. During the talk, we will study some basic properties of UMD spaces and mention several applications.

Piotr Zgliczyński

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.

Armen Edigarian

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.

Mikołaj Bojańczyk

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.

Balázs Bárány

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.

Maciej Dołęga

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.

Andrzej Nagórko

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.

Javier de Lucas

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.

Piotr Przytycki

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.

Jan Rudnik

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 $Z_2$ principal bundle over it and prove a non-triviality result about the tautological line bundle.