rok akademicki 2010/2011

27.05.2011Tadeusz Kulczycki The Steklov problem and hydrodynamics
13.05.2011Sławomir Szczepaniak Distortable Banach spaces
6.05.2011Łukasz Kubat Some ring-theoretical properties of plactic algebras
29.04.2011Tomasz Komorowski Homogenization of solutions of elliptic and parabolic partial differential equations with random coefficients
15.04.2011Paweł Strzelecki Szkic rozwiązania XIX i XX problemu Hilberta
8.04.2011Maria Donten-Bury Invariants of Kummer 3-folds
1.04.2011Ben Warhurst Carnot groups and subriemannian geometry
25.03.2011Andrzej Sitarz Noncommutative Geometry - a panoramic view
18.03.2011Jerzy Lewandowski Jak pogodzić matematykę, teorię względności i kwantową teorię pola?
11.03.2011Wojciech Kucharz Transcendental submanifolds of projective space
25.02.2011Radosław Adamczak Random matrices with independent rows/columns
21.01.2011Javier de Lucas Araujo Lie systems and superposition rules
14.01.2011Piotr Koszmider On a problem of S. Rolewicz
17.12.2010Wiesław Żelazko On the Grauert and Remmert theorem
3.12.2010Michał Jóźwikowski Double vector bundles
26.11.2010Paweł Hitczenko On random staircase tableaux
19.11.2010Michel Zinsmeister From Bieberbach to Schramm, a 20th century saga
5.11.2010Grzegorz Zwara Three fundamental theorems on representations of associative algebras
22.10.2010Paweł Wolff The slicing problem
15.10.2010Mark Pollicott Counting circles in circle packings

Tadeusz Kulczycki

Linear water-wave theory is a widely-used approach that allows to determine the frequencies and modes of free oscillations of a liquid in a container. Such oscillations exist provided the liquid's upper surface is free and, in the framework of this theory, one obtains their frequencies and modes from the Steklov problem. The Steklov problem is the classical eigenvalue problem which has been the subject of a great number of studies. We study the case when a liquid container is an axially symmetric, convex, bounded domain. We obtain results about fundamental frequency and fundamental modes of free oscillations of a liquid.

Sławomir Szczepaniak

After a short historical introduction to the distortion problem I will present a combinatorial approach (motivated by the forcing techniques) to constructing distortable Banach spaces.

Łukasz Kubat

During the seminar, I will talk briefly about the ring-theoretical properties of the plactic algebras over a field. In particular, I will focus on recently obtained results, together with prof. Jan Okniński, concerning irreducible representations, the primitive and the minimal prime spectrum and the Groebner-Shirshov bases for these algebras.

Tomasz Komorowski

In my talk I will discuss the problem of finding the limit of solutions of some types of partial differential equations with fast oscillating random coefficients. This type of problem occurs when there are two distinct, widely separated scales present (microscopic vs. macroscopic ones), e.g. in the problem of turbulent transport, finding an effective electric field in semiconductors, some chemistry problems and even in financial engineering. The solutions of equations with fast oscillating, random coefficients converge, in an appropriate sense, to a solution of an appropriate constant coefficient equation, the so called effective equation. Because the limiting procedure removes the inhomogeneity related to the description of the environment it is sometimes called the homogenization. We shall discuss both the analytic approach, via the G-convergence technique, and the probabilistic method, via the martingale central limit theorem, used in the homogenization theory.

Paweł Strzelecki

Opowiem krótko, skąd wzięło się pojęcie słabego rozwiązania równania eliptycznego. Następnie naszkicuję, w jaki sposób w końcu lat 50-tych udowodniono, że każde słabe rozwiązanie równania eliptycznego (niekoniecznie liniowego!) jest funkcją gładką. Powiem też, dlaczego dla układów równań różniczkowych cząstkowych sytuacja jest zupełnie inna niż dla pojedynczego równania. Tu, nawet dla dwóch zmiennych niezależnych i dwóch zmiennych zależnych, pytanie o to, jak rozstrzygnąć, które układy nieliniowych równań eliptycznych mają tylko rozwiązania regularne, które zaś mogą mieć także rozwiązania z osobliwościami, jest w gruncie rzeczy w pełni otwarte.

I'll tell briefly how the notion of a weak solution of an elliptic equation came into being. Then, I'll sketch the proof (dating back to late 1950's-early 1960's) of the theorem that every weak solution of an elliptic equation (possibly nonlinear) must be smooth. It turns out that for systems of elliptic equations the situation is completely different. Even for systems of two equations in two independent variables the problem of telling which system has only regular solutions and which one admits singular solutions is, basically, wide open.

Maria Donten-Bury

A Kummer surface is constructed by dividing an abelian surface by an action of an involution and resolving its 16 singular points by blowing them up once. We investigate a generalization of this construction described in a recent paper of M. Andreatta and J. A. Wiśniewski. By taking a crepant resolution of singularities of a quotient of an abelian variety by a finite integral matrix group action we obtain a generalized Kummer variety, which has trivial first cohomology and canonical class. In particular, in dimension 3 this construction gives Calabi-Yau varieties.

In this talk we will show the classification of all possible results of the generalized Kummer construction in dimension 3, which involves classifying finite subgroups of SL(3,Z). We will also describe algorithms of computing their Poincaré polynomials and fundamental groups.

Ben Warhurst

I will introduce subriemannian geometries and discuss the role of Carnot groups in their analysis.

Andrzej Sitarz

NCG (à la Connes) is the beginning of a complete program to put most (being very optimistic) or at least some (being more realistic) geometries into a very general frame: that of algebra of operators. Starting from the perspective of physics I'll describe where NCG comes from, what is its main focus and how "noncommutative spaces" are described. I'll try to briefly present the main benefits of the noncommutative geometry tools, both in mathematics (around the index theorem) as well as in mathematical physics (new approach to gauge theories and action principle).

Jerzy Lewandowski

Celem wykładu jest przedstawienie problemów matematycznych, do których prowadzi kwantowa teoria pola i teoria względności. Na wstępie podane zostanie matematyczne sformułowanie kwantowej teorii pola w zakrzywionej czasoprzestrzeni. Następnie przedstawione będą przykłady teorii kwantowej niezmienniczej ze względu na dyfeomomorfizmy. Głównym punktem wykładu jest sformułowanie matematycznego schematu grawitacji kwantowej. Wykładowca dołoży starań, aby przytoczone przykłady problemów matematycznych zainteresowały obecnych na widowni matematyków.

Wojciech Kucharz

A smooth submanifold M of Pn(R) is said to be of algebraic type if it is isotopic in Pn(R) to the set of real points of a nonsingular complex algebraic subset of Pn(C) defined over R; otherwise M is said to be transcendental. It is not at all obvious that transcendental submanifolds exist (for example any codimension one smooth subamnifold is of algebraic type). I will describe a surprizingly simple construction of transcendental submanifolds in all codimension greater than one.

Of special interest are transcendental submanifolds of codimension two. They can be explicitly characterized.

Radosław Adamczak

The classical theory of random matrices is focused on two types of models, matrices whose entries are independent random variables (possibly up to a symmetry condition) and matrices whose distribution is invariant under conjugation by orthogonal (or unitary) matrices.

In my talk I will discuss a different model of a random matrix, in which one assumes statistical independence of rows (resp. columns) of the matrix but allows the entries within a row (resp. column) to be dependent random variables. I will briefly present some problems in statistics, computer science and high dimensional geometry in which such matrices arise and indicate various conditions (usually of geometric nature) that one may impose on the rows (resp. columns) of the matrix to compensate for the lack of full independence of the entries.

Javier de Lucas Araujo

Lie systems form a special class of systems of first-order ordinary differential equations whose general solutions can be expressed in terms of certain families of particular solutions and a set of constants, by means of a special type of map: the so-called superposition rule.
Apart from this remarkable feature, Lie systems admit multiple interesting geometrical properties that can be employed to investigate diverse differential equations appearing in the physics and mathematical literature. In this talk, we intend to depict some of these geometrical properties as well as to point out a number of their applications to mathematics and physics.

Piotr Koszmider

Convex, bounded and non-empty set C in a Banach space is called a support set if every point of it is a point where some linear continuous functional assumes its proper infimum in C. S. Rolewicz showed that there are no support sets in separable Banach spaces. However, the problem if there are support sets in every non-separable Banach space was the subject of many publications and finally after 30 years it turned out that it is undecidable. I will discuss the related notions and results from applications of infinite combinatorics and logic in Banach spaces.

Wiesław Żelazko

Grauert i Remmert pokazali, że przemienna Noetherowska algebra Banacha jest skończeniewymiarowa (odwrotny fakt jest oczywisty). Wynik jest oparty na stwierdzeniu, że jeżeli ideał takiej algebry ma skończenie generowane domknięcie, to sam musi być domknięty.

W swoim wykładzie pokażę, że można poprawić twierdzenie Grauerta-Remmerta i zakładać jedynie, że ideały maksymalne są skończenie generowane (algebraicznie), oraz pokażę, że dla algebr środkowych stwierdzenie odwrotne do powyższego jest również prawdziwe.

Jeśli czas pozwoli, opowiem o dalszych uogólnieniach twierdzenia Grauerta-Remmerta.

Michał Jóźwikowski

A double vector bundle (DVB) is a manifold equipped with two vector bundle structures which satisfy certain compatibility conditions. Natural examples of such objects are tangent and cotangent bundles of a vector bundle. During the lecture I shall discuss a few examples to illustrate the basic concepts of DVB theory. I will also show how the language of DVBs can be used in geometric mechanics and theory of connections.

Paweł Hitczenko

We discuss a new combinatorial structure, called staircase tableaux that was introduced in 2009 by S. Corteel and L. Williams in connection with the Asymmetric Exclusion Process (ASEP), a particle model used in statistical mechanics. Subsequently, Corteel and Williams called for an investigation of basic properties of staircase tableaux. In this talk we take a first step in that direction: after defining staircase tableaux and briefly describing their connection to ASEP we will describe a probabilistic approach that allows us to analyze some basic parameters of randomly chosen staircase tableau of a given size.

This talk is based on joint work with Sandrine Dasse-Hartaut (currently a PhD student at LIAFA).

Michel Zinsmeister

The famous Bieberbach conjecture was stated in 1916. The first breakthrough towards its proof was done by Loewner in 1923. Loewner method have been revived in a totally different framework by Schramm in 1999. The aim of this talk is to revisit Bieberbach coefficient problem and the steps of its proof in the framework of Schramm SLE processes.

Grzegorz Zwara

Let k be an algebraically closed field. Let A be an associative k-algebra with an identity, which is finite dimensional (as a vector space over k). We are interested in the category of finite dimensional left A-modules. This reduces to study the indecomposable modules and the homomorphisms between them. At the origin of the present developments of the representation theory (in the nineteen seventies) is the introduction of bound quivers by Gabriel, and on the other hand, of almost split sequences by Auslander and Reiten. The third result to be discussed during my talk is the theorem due to Drozd that the finite dimensional k-algebras form two disjoint classes: of tame and of wild representation type.

Paweł Wolff

The slicing problem (a.k.a. the hyperplane conjecture) asks whether there exists a universal constant c>0 such that in any dimension n, any convex body of volume 1 in Rⁿ has at least one hyperplane section whose (n-1)-dimensional volume is >c. Despite much attention over the past 25 years, this question still remains open.

I will give a brief overview of the history of the question and discuss some related problems. I will also highlight some tools and ideas that have been used to establish various partial results.

Mark Pollicott

The Apollonian circle packing consists of infinitely circles generated from triples of mutually tangent circles and the limit set is a well known fractal. There are interesting results on the sequences of real number which occur as radii of the circles for both this case, and related examples.