Lyapunov exponents tell us the rate of divergence of nearby trajectories – a key component of chaotic dynamics. For one dimensional maps Lyapunov exponent at a point is a Birkhoff average of $\log|f'|$ along the trajectory of this point. For a typical point for an ergodic invariant measure it is equal to the average of $\log|f'|$ with respect to this measure.

In this talk, we will give an introduction about Lyapunov exponents of cocycles. Moreover we will give a few examples which are related to PDE, Smooth Dynamics and Probability. We deal with Lyapunov exponents of products of random i.i.d. $2\times2$ matrices of determinant $\pm1$. We will see how Lyapunov exponent gives information about the norm of growth of the matrices $A^{n}(x)$. Finally we will discuss conditions in which the Lyapunov exponents are always positive.

Energies of submanifolds are an analytic tool for studying topological and geometric properties of embedded submanifolds. Interest in energies was rekindled in the nineties by pioneering works of Freedman, of He and Wang and of O'Hara, who used energies of curves to study knottedness of links in R³. One of the most useful energies is the Menger curvature defined originally for curves in R³, and which was studied in detail by many authors. The Menger curvature can be generalized to surfaces and to higher-dimensional submanifolds. The intuition behind the energy is that, if a submanifold has complicated topological behavior (for example the curve is knotted), then the energy should be large. In other words, small energy should imply simple topology. We want to look at some basic definitions and intuitions in this field. For instance, we shall review some properties of energies, and discuss difficulties and unexpected traps one encounters while trying to extend the definition of energy to higher dimensions. At the end, we will sketch a proof that there exists a positive constant C such that, for every smooth, closed and connected surface S in R³ of total area 1, the genus g(S) and the energy E(S) of the surface S satisfy the inequality: g(S) less than or equal to C times E(S).

On a warm sunny Friday afternoon in Warsaw, we shall take a close look at some of the key examples of C*-algebras. In the process, we will learn a bit about the general theory of C*-algebras and why they are worth studying. I will then speak briefly about the dramatic recent progress in the classification of C*-algebras and my research in this area.

The first part of the presentation is a colloquium-style introduction to the mathematics of quasicrystals as seen from the theoretical physicist point of view. We will discuss simple example of optimization problems in kissing numbers, packing spheres, non-periodic tilings, and classical-lattice gas models of statistical mechanics. We will also formulate the main open problem - the existence of non-periodic Gibbs measures for finite-range hamiltonians.

In the second part of the talk, we will discuss various connections between ergodic theory, substitution dynamics, dynamical systems of finite type, and classical lattice gas-models.

We will outline a research project supported by NCN Harmonia grant "Mathematical models of quasicrystals". We are looking for young people to collaborate within the framework of the grant.

I will discuss functions possessing the mean value property. Such class has been introduced in the metric measure spaces as one of possible definitions of harmonic functions in this setting. During the talk I will focus on the consequences which the property supports and present a large collection of examples to complete the analysis.

I will talk about the Gromov density model, in which a group is obtained by specifying generators and then imposing a random set of relations between them. The model is simple, yet produces groups with interesting geometric properties and provides testing grounds for various methods and conjectures. I will describe some of the landscape, focusing later on left-orderability and property (T) (which leads to a connection with random graphs). No previous knowledge of the topic(s) is assumed.

In this talk we will give an introduction to the iteration of transcendental entire functions focusing on the escaping set. The escaping set consists of points that go to infinity under iteration and it plays an important role in the area. After giving the definition and discussing some of its properties, we will look at the structure of the escaping set. In particular, we will present two specific examples of transcendental entire functions for which the structure of the escaping set differs significantly.

C*-algebras can be viewed as operator norm-closed self-adjoint subalgebras of bounded operators on Hilbert spaces. Because of this, the theory of C*-algebras benefits from a rich representation theory. As the kernel of a representation is a norm closed two-sided ideal of a C*-algebra, a major tool in this theory comes in the form of topologies on ideals, where ideals are seen as points. This begins with the Jacobson topology on primitive ideals, which are ideals formed by the kernels of a non-zero irreducible representations. From this, J.M.G. Fell developed a topology on all norm closed two-sided ideals of a C*-algebra. Motivated in part by this, we developed a new topology on the ideal spaces of C*-algebras formed by inductive limits (C*-inductive limits). We then compare this topology to the other topologies discussed. We also present that our topology does agree with Fell's topology for the particular C*-inductive limits called approximately finite-dimensional C*-algebras (AF-algebras). As an application, we first provide a metric topology on certain quotients of AF-algebras using the tools of Noncommutative Metric Geometry, in particular M.A. Rieffel's compact quantum metric spaces and F. Latremoliere's quantum Gromov-Hausdorff propinquity. Next, we introduce sufficient conditions for when our topology on ideals produces a continuous map from certain sets of ideals to the associated space of quotients. An example of such a continuous map is given by the Boca-Mundici AF-algebras. This shows that the act of taking a quotient can be seen to be continuous at the level of viewing ideals and quotients as points of topological spaces.

The notion of Szlenk index was introduced in 1968 by W. Szlenk in order to show that there is no universal Banach space in the class of all separable reflexive Banach spaces. Its origins stem from the Cantor-Bendixson index which is well-known in topology. Since the pioneering paper by Szlenk, his index and several similar ordinal indices have proven to be extremely useful tools in Banach space theory. During the talk, we will discuss some intuitions behind the Szlenk index and the Szlenk power type, and some of the most recent results on this topic. These involve connections with asymptotic geometry of Banach spaces and the theory of asymptotic structures, which is quite fundamental for understanding the current state of knowledge in the structural theory of Banach spaces.

I will talk about directed spaces; these are topological spaces with some additional structure, which can be used for modelling concurrent programs. I will define directed counterparts of classical topological invariants and present main problems which are investigated in this area.

Do you feel that you are going around in circles and not getting anywhere? Things may not be as bad as they seem. You might be getting somewhere, but not realizing it because you aren't aware of your personal monodromy*.

In this lecture, I will provide a gentle overview of the concept of monodromy in the context of algebraic geometry, algebraic topology, differential equations, and number theory.

I will give a friendly introduction to the theory of formal group laws focusing on arithmetic questions such as integrality and local invariants.

It is well known that every $C^*$-algebra has an increasing approximate unit w.r.t. the usual partial order on the positive unit ball. We consider the strict order $<<$ instead, where $a << b$ means $a = ab$. Here again it is well known that every separable or sigma-unital $C^*$-algebra has a $<<$-increasing approximate unit, but the general case remained unresolved. In this talk we outline our recent work showing that this extends to $\omega_1$-unital $C^*$-algebras but not, in general, to $\omega_2$-unital $C^*$-algebras. In particular, we consider $C^*$-algebras defined from Kurepa/Canadian trees which are scattered and hence LF but not AF in the sense of Farah and Katsura. It follows that whether all separably representable LF-algebras are AF is independent of ZFC.

For a locally compact group $G$, let $C_{b}(G)$ be the space of all complex-valued, continuous and bounded functions on $G$ equipped with the sup-norm, and $LUC(G)$ be the subspace of $C_{b}(G)$ consisting of all functions $f$ such that the map $G\to C_b(G);x\mapsto l_xf$ is continuous, where $l_xf$ is the function defined by $l_xf(y)=f(xy)$, for each $y\in G$. The subspace $LUC(G)$ forms a unital commutative C*-algebra. We can induce a multiplication on the Gelfand spectrum of $LUC(G)$, $G^{LUC}$, with which $G^{LUC}$ forms a semigroup. In this talk, I study some properties of $G^{LUC}$, the so called right topological semigroup compactification of $G$. I also discuss the question of when the corona, $G^{LUC}\setminus G$, determines the underlying topological group $G$.

I will discuss two classical probabilistic objects - random walks and birth-death processes - using the language of operator theory. We will see how commutation relations between certain operators (related to the generators of the aforementioned processes) allow us to perform explicit computations; the combinatorial tools from free probability, such as non-crossing partitions, will appear naturally. If time permits, I will show how to make a transition from classical probability to quantum probability.
P.S. I may present an example involving Darth Vader and stormtroopers.