Opowiem o klasycznym problemie, jak rozstrzygnąć, czy dana koneksja jest koneksją Levi-Civity i ogólniej, czy klasa projektywnie równoważnych koneksji zawiera koneksje Levi-Civity (dwie koneksje są projektywnie równoważne, gdy mają te same geodezyjne). W rozwiązaniu problemu kluczowa jest analiza grupy holonomii.

Mother Nature seldom produces ideal materials. For example ideal semiconducting crystals will be of little use and we made them "dirty" for purpose. There is one particular class of imperfections in matter which we call topological defects. When you watch this text on LCD display of a laptop you are seeing a lots of them. In this lecture I would outline results of our research on a particular class of such a topological defects occurring in solids. It turns out that the mathematical tools used there are those provided by differential geometry of manifolds with both curvature and torsion. Torsion plays here the major role and is a directly measurable physical quantity. It turns out that all that has close relation to the important effect discovered in XX century in quantum mechanics called Aharonov-Bohm effect.

The problem of reconstruction of semi-algebraic sets and functions from integral measurements, like moments or Fourier transform, naturally arises in Signal Processing on one side and in Qualitative Theory of ODE's on the other. Because of the inherent discontinuity of the data our reconstruction approach is algebraic and non-linear. I plan to stress Signal Processing and Approximation Theory aspects of the problem ("Algebraic Sampling"). In particular, I plan to discuss the optimal accuracy of reconstruction of piecewise smooth functions from a given number of their Fourier coefficients. This problem has been open for a pretty long time. Recently in a joint work of D. Batenkov and myself a partial answer has been found, which may turn out also to be a complete one.

If time allows, I plan to discuss also the uniqueness aspects of reconstruction. This leads to the moment vanishing problem: give conditions for identical vanishing of the moments m_k=∫P^k(x)q(x)dx, for various classes of P and q, and various integration domains. Recently a serious progress has been achieved in some special cases of this problem, and relations have been found with the Mathieu conjecture in representations of compact Lie groups, and (through the recent work of Wenhua Zhao) with certain questions around the Jacobian conjecture.

I will present a few fundamental facts and questions on Abelian varieties and elliptic curves defined over number fields (e.g., the problem of computing ranks of Mordell-Weil groups will be discussed). In the second part of the talk we will focus on representations of Galois groups on torsion points and explain why the representations may be useful in the arithmetics. This is going to be an expository lecture for general audience of graduate students in mathematics (only a basic training in algebra or in number theory required).

I will discuss how one can use signatures of torus knots, so a topological link invariant to study properties of plane algebraic curves and deformations of plane curve singularities.

Rozważmy obszar D oraz punkt p leżący na jego brzegu. Przypuśćmy, że nieujemne funkcje f i g są harmoniczne w D, ciągłe na brzegu D, zaś w pewnym otoczeniu punktu p są równe zeru na brzegu D. Brzegowa nierówność Harnacka (ang. BHI) mówi o porównywalności f i g w pobliżu p: jeśli tylko brzeg zbioru D jest dostatecznie regularny, iloraz f/g jest w przybliżeniu stały w otoczeniu p.

Motywacją do badania tego typu zagadnień było przede wszystkim zastosowanie funkcji harmonicznych w fizyce, w teorii potencjału. Wraz z rozwojem tej teorii zaczęto rozważać funkcje harmoniczne względem innych operatorów niż laplasjan, lub równoważnie (w probabilistycznej interpretacji teorii potencjału) – względem innych procesów niż ruch Browna. Mnie szczególnie interesują operatory nielokalne (w języku probabilistycznym – procesy skokowe), ze szczególnym uwzględnieniem ułamkowego laplasjanu (któremu odpowiada symetryczny proces stabilny).

Zaskakujące jest to, że wiele twierdzeń klasycznej teorii potencjału można udowodnić dla operatorów nielokalnych. Jeszcze ciekawsze jest to, że niektóre wyniki, w tym BHI, w przypadku nielokalnym mogą być prawdziwe bez założeń o geometrii brzegu.

W czasie referatu przedstawię krótkie wprowadzenie do teorii potencjału w ujęciu analitycznym i probabilistycznym. Podam również elementarny dowód BHI w klasycznym przypadku dla zbiorów o gładkim brzegu oraz naszkicuję metodę dowodu BHI dla procesów skokowych.

Consider a domain D and its boundary point p. Suppose that nonnegative functions f and g are harmonic in D, continuous at the boundary of D and vanish at a part of the boundary in a neighborhood of p. The Boundary Harnack Inequality (BHI) states that in this case f and g are comparable near p, provided that the boundary of D is sufficiently regular.

The study of boundary behavior of harmonic functions was motivated by numerous applications of the concept of harmonic functions to physics. For example, (electromagnetic and gravitational) potentials are described by harmonic functions. In the mathematical potential theory some generalizations are also studied, for example, functions harmonic with respect to operators other than Laplacian, or, in the probabilistic approach, with respect to Markov processes other than Brownian motion. I am particularly interested in the nonlocal operator case, which corresponds to jump processes. The most important example here are α-harmonic functions, that is, functions harmonic with respect to the fractional Laplacian (or the symmetric stable process).

It may be surprising that many results of the classical potential theory generalize to the nonlocal case. And it is even more striking that in the nonlocal setting some theorems, including BHI, hold true with no regularity conditions on the boundary of D.

I would like to start my talk with a short introduction to the analytic and probabilistic approach to potential theory. I will give an elementary proof of BHI in the classical case for domains with smooth boundary and sketch a method of proving BHI for jump processes.

I will present some basic facts about measuring the expected return and risk connected with assets from a Stock Exchange. I will show the Harry Markowitz' approach to finding the optimal asset allocation, based on Kuhn-Tucker theorem.

I will present some of the well known in biology models of evolution and their mathematical properties. I will show how one can construct toric varieties associated to these models. I will describe properties of these varieties and say what are the possible applications of this approach.

The talk includes

1. Old and new in iteration theory
2. Asymptotic behavior of one-parameter semigroups
3. Rigidity of holomorphic functions
4. Linearization models and dynamic approach to geometric function theory

Young measure theory is one of the fundamental tools in the modern Calculus of Variations.
It describes (weak) limits of sequences of compositions {f(u^j)} under the assumption that the sequence {u^j} is bounded in some Banach space, for example in L^p. The purpose of the lecture is to present the brief introduction to the Young measure theory, to explain its effectiveness in relation to the classical problems, as well as to discuss some important problems in Young measure theory, which arise from modern Calculus of Variations.

We present an area-type formula to compute the intrinsic measure of submanifolds immersed in a stratified group equipped with a sub-Riemannian metric structure.

In 1955 C. Chevalley showed how to construct analogies of the complex semi-simple Lie groups over arbitrary fields. Nowadays, these groups are often called "Chevalley groups" or "Groups of Lie type". He also realized that after an appropriate reformulation, many theorems about semisimple Lie groups admit analogues for Chevalley groups. Moreover, simple Chevalley groups over finite fields were the starting point in the classification of finite simple groups.

During the talk I will give a survey about Chevalley groups. Many properties of Chevalley groups (independent of the field of definition) can be clearly presented and proved in terms of the Weyl group, which is an example of Coxeter group. Hence, my talk will be complementary to P. Przytycki's lecture on Coxeter groups, presented in December 2009.

I will talk about a few selected facts from asymptotic convex geometry. My aim will be to show the sort of problems one may be interested in around this subject, along with a few of the elementary techniques used to solve them. I will, in particular, show a "far away from zero" concentration of measure phenomenon for permutation invariant balls (along with explaining what a concentration of measure phenomenon is, what are permutation invariant balls, and why is the whole problem interesting). I will also talk about a few other facts concerning permutation invariant balls. I will pose a few open problems, which look accessible and are formulated in an understandable language.

Let f be a continuous map of a closed interval to itself or an orientation preserving self-homeomorphism of a closed disk. Periodic orbits of f can be classified according to their properties, like their (minimal) periods or permutations in one dimension, or their braid types (that tell us how, up to isotopy, the map acts in the complement of the periodic orbit) in two dimensions. This defines a type of the orbit. I will discuss the forcing relation among those types: when the existence of an orbit of one type implies the existence of a periodic orbit of another type.

A talk for beginners on what are Coxeter groups, what are the most popular examples, what spaces do they act on, and what you can do with them.

Motivated by the applications for compression and study of subjective quality of the sound, we investigate the diffusion models of sound propagation in the enclosures composed of different materials. The diffusion model embraces the effect of sound delay when propagating through the material and performs well in case of reverberation times that are important properties of the enclosure. The computational complexity of the method is reasonable when compared with the state-of-art.

Monoid chiński rangi n to półgrupa z jedynką, zdefiniowana (poprzez podanie generatorów i relacji) w następujący sposób:

W swoim referacie postaram się opowiedzieć, jakie motywacje mogą skłaniać do badania tego typu obiektów, jakimi metodami próbowałam je badać oraz jakiego typu rezultaty udało się otrzymać w mojej rozprawie doktorskiej.

I will talk about an axiomatic theory known as Peano Arithmetic (PA). PA was originally intended to be about the natural numbers, but it can be viewed as the analogue of Zermelo-Fraenkel set theory describing the realm of "finite mathematical objects".

PA is a surprisingly powerful theory: it proves almost all known mathematical theorems expressible in its language (which is a much weaker restriction than might seem at first sight). However, there are some nontrivial examples of true statements known to be unprovable in PA. One such statement is the so-called Paris-Harrington theorem, a generalization of the finite Ramsey theorem. I will discuss this example and, if time permits, try to sketch the proof of unprovability.

Przedstawię jedno ze zjawisk przewidywanych przez ogólną teorię względności, mianowicie fale grawitacyjne oraz metody wykrywania tych fal. Ze względu na to, że amplitudy fal grawitacyjnych są niezwykle małe, ich wykrycie jest ogromnym wyzwaniem dla statystycznych metod detekcji sygnałów. Przedstawię szereg matematycznych narzędzi i algorytmów numerycznych, które optymalizują problem detekcji sygnałów fal grawitacyjnych i estymacji ich parametrów. Zaprezentuję zastosowanie tych metod do analizy rzeczywistych danych z działających obecnie detektorów fal grawitacyjnych.

I will talk about a problem which I was dealing with at the school on vector bundles PRAGMATIC in September. I will construct a full, strongly exceptional collection of line bundles in the derived category of some smooth, complete toric varieties with Picard number three. As the main tool I will use the splitting of the Frobenius direct image of line bundles on toric varieties, as well as a precise combinatorial description of acyclic line bundles.

Let M be a flat manifold, that is compact Riemannian manifold with vanishing curvature tensor. Differentiable type of M is determined by its fundamental group, which is a torsionfree crystallographic group. Assume that additionally M is a complex manifold and flat metric is Kaehler. I will state some characterisation of such manifolds in terms of its fundamental groups. Then I will present classification results in low dimensions. I will also give a detailed description of a class of flat Calabi-Yau manifolds, which is a generalisation of a class of Hantzsche-Wendt manifolds.

I shall talk about curves in the real plane R², but many ideas work also for higher dimensions (possibly I shall mention one or two). Often I shall allow affine transformations of the curves, not just euclidean. Here are some of the questions that will probably come into the talk. Given a (say convex) simple closed plane curve C, let us draw chords (segments joining two points of C) to cut off a fixed area A inside the curve. What is the envelope of such chords? (For example, for C an equilateral triangle, A = half the area of the triangle.) What is the locus of the midpoints of such chords? As we vary A how does the envelope of chords vary? Do the singular points (cusps) of the envelope trace out an interesting curve? (Perhaps this is similar to ordinary parallels of a plane curve in euclidean differential geometry?) Given a point P inside C how many chords have P as their midpoint? How can we measure how close C is to having a centre of symmetry? Are there other affinely symmetric measures of 'symmetry' or 'local symmetry'?

I will present the concept of Borel summability of formal power series and will discuss its application to power series solutions of partial differential equations. I will characterize the summable formal solutions of linear partial differential equation with constant coefficients in terms of analytic continuation of the initial data.

Opowiem, jak definiować dla krzywych i powierzchni (w przestrzeni Euklidesowej) funkcjonały wariacyjne, które są całkowymi odpowiednikami krzywizny. Ze skończoności takich funkcjonałów można stosunkowo prosto czerpać informacje dwojakiego rodzaju: o podwyższonej regularności krzywej bądź powierzchni, a także o braku samoprzecięć. Dzięki temu można uzyskiwać twierdzenia o zwartości rodzin krzywych (powierzchni) oraz wyciągać wnioski natury topologicznej (np. dotyczące klasy zawęźlenia).