I will start from a stupid question: "what is the second derivative?". An answer given from the point of view of differential geometry will help us to discuss second order optimality conditions in variational calculus. As an example we will consider the geodesic problem and related issues – the Jacobi equation and Morse Theory.

Variational calculus and second derivatives(joint paper with Arkadiusz Pawlik, Jakub Kozik, Tomasz Krawczyk, Piotr Micek, William T. Trotter, and Bartosz Walczak) In 1970s Erdos asked whether the chromatic number of intersection graphs of line segments in the plane was bounded in terms of the size of the largest clique. We show the answer is no. Furthermore, the very same graphs can be represented by intersections of other planar objects like axis-alligned rectangular frames or L-shapes. The latter provides a negative answer to a question of Gyarfas and Lehel (1985). Our construction disproves also a purely graph-theoretical conjecture of Scott (1997).

Let X be a random vector taking values in a Banach space with a norm ║·║. Studying various aspects such as Central Limit Theorems or Large Deviations it is natural to ask about precise estimates on the probability of the event {║X║>t}, t>0. In the talk we will try to find the supremum of the quantity P(║X║>t) when the norm ║·║ changes. The relevant inequalities are called S-inequalities. We will address the cases of basic distributions, e.g. Gaussian, exponential.

Describing isomorphism classes of varieties is very difficult in general. One possible approach is to start classifying varieties up to birational equivalence, i.e., identifying varieties that have isomorphic Zariski open sets. One of the main achievement of the Italian algebraic geometry school at the beginning of the last century was the birational classification of complex projective surfaces, that is: to find a good representative (a so-called minimal model) in any fixed birational equivalence class. It turned out that while for surfaces the recipe is quite simple, for varieties of higher dimension some major problems arise. Anyway since the early 1980s the works of Reid and Mori raised the hope that a higher dimensional analogue could be possible. In this talk I discuss the framework of the Minimal Model Program and one of its main machinery, the cone theorem.

In my talk I will introduce Euler equations giving (not very rigorous) physical meaning of the terms appearing there. Next I will try to convince the audience that considering 2d model as well as less regular solutions to the Euler equations makes sense from the point of view of physics and engineering. After proving uniqueness of classical solutions I will pass to the famous Yudovich theorem on uniqueness of vortex patches. Next, after introducing the concept of vortex sheet, I will discuss the open problems of (non-)uniqueness of both DiPerna-Majda solutions and vortex sheets. At the end of the talk I will sketch a new proof of the Yudovich theorem following the ideas of Witold Wolibner.

In the breakthrough paper A new measure conjugacy invariant for actions of free groups, Ann. of Math. 171 (2010), 1387–1400, Lewis Bowen solved a long standing problem in measurable dynamics, namely the classification of Bernoulli shifts over free group of finite rank up to isomorphism. This was achieved by a developing a theory of „f-entropy” for measure preserving free group actions. In my talk I will present the key elements of the elegant yet elementary construction of the „f-(entropy) invariant”. If time permits I will also discuss my joint work with Lewis Bowen where we generalized the Rokhlin-Abramov formula and the Juzvinskii Addition Theorem to the setting of free group actions. The talk will be completely self contained and all concepts will be defined, however it may be helpful to read the following lines:

A ($\mathbb{Z}$-)measure preserving system consists of a probability space $(X,B,\mu)$ ($X$ is a set, $B$ is a sigma-algebra, $\mu$ is a probability measure) and a bi-measurable map $T:X\to X$ so that for any measurable set $A$, $\mu(TA)=\mu(A)$. A partition $\alpha=\{A_1,\dots,A_n\}$ is a pairwise disjoint collection of measurable subsets $A_i$ of $X$ whose union is $X$. If $\alpha$ and $\beta$ are partitions then their join consists of all possible intersections of elements of $\alpha$ and $\beta$. One says $\alpha$ refines $\beta$ if the join of $\alpha$ and $\beta$ equals $\alpha$, in other words if each element of $\alpha$ is contained in an element of $\beta$ (up to measure 0). Let $K$ be a finite set and $\kappa$ a probability measure on $K$. Let $K^{\mathbb{Z}}$ be the product space with the product measure $\kappa^{\mathbb{Z}}$. The measure preserving system consisting of the probability space ($K^{\mathbb{Z}},\kappa^{\mathbb{Z}}$) and the shift transformation $K^{\mathbb{Z}}\to K^{\mathbb{Z}}$ given by $T(x)(n)=x(n-1)$ is called the Bernoulli shift.

Rank of a homogeneous polynomial f of degree d is the minimal number k such that there exist linear forms l₁, ... l_k with f=l₁^d+…+l_k^d. The problem of finding the rank of a polynomial in two variables was solved by Sylvester in XIX century. I will summarise his result. For more variables it is still open, but bounds are known.

More generally one can ask about rank of a tensor. Matrix multiplication is a particularly interesting case.

Since the publication in late fifties of the first articles on the topic, the theory of random graphs has developed quickly and found many applications. In my talk I will introduce briefly the history of random graphs. I will present origins of the field and some of the most intriguing and fertile problems concerning random graph models. At the end I will mention some recent results and their applications.

Idempotent probability measures on a (locally) compact group G arise naturally as limit distributions for random walks on G. A classical result of Kawada and Ito characterises them as Haar measures on compact subgroups of G. We will discuss this result, its dual version for 0-1 valued positive-definite functions on discrete groups and then present a natural `quantum' generalization of the concept of an idempotent measure: the notion of an idempotent state on a compact quantum group. It turns out that although (or because!) the Kawada-Ito theorem does not hold in the quantum world, considering idempotent states leads to an interesting and rich theory. We will also mention possible generalizations of idempotent measures in other directions, related to some questions of classical and noncommutative harmonic analysis.

The class of regular languages of words over a finite alphabet A can be defined in many equivalent ways:

1. the closure of the family of singletons under finite union, concatenation and Kleene star (X* is the set of concatenations of arbitrary many words from X);
2. languages definable in monadic second order logic;
3. languages recognized by finite automata;
4. preimages under homomorphisms from A* into finite monoids.

In my talk I will consider first order quadratic differential equations, i.e. Riccati equations, which in general cannot be solved by quadratures. I will show a well known connection of such equations with Lie-Scheffers systems on the group SL(2), and then writing solutions to the latter systems by infinite quadratures, I will obtain the same for general Riccati equations.

A tiling of, say, the Euclidean plane is a tessellation, in which every subset is a polytope, isometric to one of finitely many prototiles. Such a tiling is aperiodic if no shifted copy will overlap with the original. We will be interested in finite sets of prototiles, with the property that every tiling by these tiles is aperiodic. Such tiles were first constructed in 1966 by Berger for the Euclidean plane, famous examples are Penrose's “kites” and “darts”. There are also constructions of aperiodic tiles for other non-compact manifolds such as the hyperbolic plane or certain symmetric spaces.

Aperiodic tilings are a mathematical model of quasi-crystals, solids, whose X-ray diffraction patterns show discreteness similar to crystals, but whose symmetries are incompatible with crystal structure.

In this talk we will address the question of existence of aperiodic tiles for various non-compact manifolds and show how one can employ large scale geometry to construct such tiles in certain cases, in particular in the case of 3-dimensional geometries appearing in Thurston's geometrization conjecture. This talk is based on recent joint work with Shmuel Weinberger.