A C*- algebra is an algebra of operators  on a Hilbert space which is closed in norm and invariant under taking adjoints. It is simple if it does not have any non-trivial ideals.  We discuss some well known examples and ways to decide if they are isomorphic or not. This uses a very deep and powerful machinery  of a K-theoretic nature which was developed originally  for C*-algebras  but has far reaching influence and applications in other mathematical fields such as algebraic topology, differential geometry or even algebraic geometry.

The discussion will take place after Professor Bartosz Naskręcki's lecture. I would really like the students to do most of the talking. We will speak English so that the discussion is accessible to everybody, but contributions in Polish are perfectly fine. There are no wrong questions or comments and all your remarks are welcome. We may start with questions for Professor Naskręcki or share our own experiences with AI. Every participant may prepare a presentation of one or two slides to explain their experience more clearly.

The discussion can be held on two levels: a philosophical one, concerned with the global picture and possible future directions, and a practical one, concerned with how to develop a new style of doing mathematics while remaining part of traditional research.

A list of questions to inspire participants has been put together by me and has been distributed among them. If anyone would like to contact me about anything before the discussion, I would be delighted to hear from them.

In this talk, we will explore the fascinating world of aperiodic tilings. We will start with some basic notions and elementary examples, such as one dimensional dominos tiling and Fibonacci tilings, and then move to more elaborate examples of aperiodic tilings of the Euclidean plane. The main examples are Wang tilings, whose decidability problem started the interest in aperiodic protosets, and the celebrated Penrose tilings which are admired in public architectural installations, university courtyards, and historic buildings around the world.

Using card shuffling as our main example, we will discuss Markov chains that exhibit an abrupt transition to equilibrium. Instead of decreasing gradually over time, their distance to equilibrium stays close to its maximal value for a long period and then rapidly drops to zero around a critical time.

Set theory provides a convenient foundation for mathematics, providing much freedom and expressive power. However, this freedom causes the existence of pathological objects like nowhere differentiable functions or nonmeasurable sets.

Model theory suggests a different route: one should restrict attention to sets that are "definable" in a suitable way (depending on the desired applications). For this approach to work, a few things have to be satisfied:

1) Definable sets are sufficiently tame to allow for some sort of structure theory.

2) The class of definable sets should be expressible enough to capture the objects of interest.

A striking example of the above approach is o-minimality, which in its basic form can be considered a "tame definable variant of analysis". One can express a lot of mathematics in this framework. The calculus developed in this world has nice properties, e.g. every function defined on an interval is differentiable everywhere except at finitely many points!

There are powerful tools in o-minimality, some of which will be discussed in this talk. The Pila-Wilkie theorem provides a deep understanding of rational points on various analytically defined sets. This has huge applications in Diophantine geometry - the Fields Medalist Jacob Tsimerman together with his coauthors was able to prove a deep conjecture (the André-Oort conjecture) using o-minimality. I will try to give a tour of o-minimality and how it applies to number theory. 

A family F of infinite subsets of the natural numbers is called almost disjoint if the intersection of any two distinct elements of F is finite. Uncountable almost disjoint families can be obtained by considering sequences of rational numbers converging to distinct reals and then fixing a bijection between the rationals and the natural numbers.

There are almost disjoint families with a wide variety of properties, usually constructed using set-theoretic methods. In this talk, however, I will focus on the use of almost disjoint families in constructions of interesting mathematical structures arising in surprisingly diverse areas of mathematics, such as topology, C*-algebras, Banach spaces or others. During the talk we shall try to discern how the combinatorial properties of the almost disjoint family manifest themselves in the resulting structures.

By basic linear algebra, the matrix AB−BA has trace 0 (i.e., is traceless), for any two square matrices A and B, of the same size, over the reals. Is every traceless matrix of the form AB−BA?

That innocuous-looking question gave rise to a century of deep results in linear algebra and beyond, starting with the Shoda/Albert/Muckenhoupt theorem, which answers it. I will discuss this result, as well as various attempts to generalize it, connections with other fields, and related still-open questions.

In this talk I want to address the fascinating interface between modern proof-based mathematics and the methods of machine learning known as large language models (simply abbreviated to GenAI). I will explain how the passage from informal mathematics to mathematical formalization, automated theorem proving and training of the GenAI transformer based models got humanity to seeing a cornucopia of results in theoretical math being proved with GenAI. I want to emphasize the current limitations, issues of training of such reasoning GenAI models and the more formal perspective on the transformer from the point of view of the possible calculations. We will play with a toy example of simply trained Qwen 1.7B base model on the rudiments of Heyting arithmetic and see how "intelligence" is born.

In this talk, I will present the recent results regarding finding the optimal upper bound for the number of paths amongst acyclic, connected graphs with N edges. I will sketch the procedure of finding the graphs that realize these bounds and show how to adapt these methods to find an optimal bound for Leavitt path algebras of a finite, acyclic, connected graph with N edges. The talk is based on joint work with P. M. Hajac, Ł. Kaczmarczyk, and M. Lowiel.

We consider the perturbed counterpart of a cellular automaton F, obtained by independently modifying each cell with probability ε and selecting a new value uniformly at random after each iteration of F. We denote the perturbed cellular automaton with noise parameter ε by F_ε. We consider two natural questions:

• For which set of parameters does F_ε admit a unique invariant measure?

• Which set of invariant measures are selected when ε goes to 0?

We will address these questions in the context of specific examples, aiming to describe the possible sets that can be reached.

Probabilistic cellular automata (PCA), introduced in the 1970s, are stochastic models used for studying how global behaviour emerges from the local interactions of many simple components. Together with their continuous-time cousins, PCA provide a framework for investigating non-equilibrium phenomena in statistical physics. PCA arising as random perturbations of deterministic CA have also been employed to study the reliability of computation in the presence of noise.

This talk will be an introduction to PCA with a focus on the ergodicity question: Does the system eventually forget its initial configuration, or can information persist indefinitely? I will present examples of both ergodicity and non-ergodicity, in particular, PCA that admit multiple macroscopic phases. Some ideas used to prove ergodicity and non-ergodicity will be sketched, and several open problems will be discussed.

This talk is an introduction to the study of surfaces and their symmetries. We will begin with the familiar group SL(2,Z) and see how it can be understood from several different points of view. These examples will lead naturally to mapping class groups and Teichmüller space, two central objects in the study of surfaces. The goal is to give a broad sense of the landscape, highlight some of the main ideas, and point toward further directions.

I will start by presenting the basic language of quantum information theory. Studying zero-error capacity of quantum channels naturally leads to the notion of quantum graphs. I will discuss the basics of quantum graphs, both the things that can be generalized from the classical case and the crucial differences. In the end I will try to connect back to quantum information theory via the notion of a quantum isomorphism.

We study sigma-ideals on Polish spaces, with particular emphasis on the Cantor space and the Baire space. Our main goal is to present useful descriptions of base sets for several classical sigma-ideals and to illustrate how such descriptions can be used in applications.

We begin with the familiar examples of the ideals of meager and null sets on the Cantor space, recalling classical results and descriptions of their base sets. We then move to the Baire space, where analogous questions lead to a richer variety of phenomena.

The final part of the talk will focus on applications. In particular, we will see how descriptions of base sets for suitable sigma-ideals can be used in results related to Mycielski-type theorems and Eggleston-type theorems.