## Referaty i plakaty

### The $\mathbb{\mathscr{NF}}$- Number of a Simplicial Complexes

Hafiz Muhammad Bilal - University of Central Punjab - Pakistan

Let $\Delta$ be a simplicial complex on $[n]$. The $\mathscr{NF}$-complex of $\Delta$ is the simplicial complex $\delta_{\mathscr{NF}}(\Delta$) on $[n]$ for which the facet ideal of $\Delta$ is equal to the Stanley-Reisner ideal of $\delta_{\mathscr{NF}}(\Delta$). Furthermore, for each $k = 2, 3, \ldots $, we introduce $k^{th}$ $\mathscr{NF}$-complex $\delta^{(k)}_{\mathscr{NF}}(\Delta)$ which is inductively defined by $\delta^{(k)}_{\mathscr{NF}}(\Delta)=\delta_{\mathscr{NF}}(\delta^{(k-1)}_{\mathscr{NF}} (\Delta))$ with setting $\delta^{(1)}_{\mathscr{NF}}(\Delta)=\delta_{\mathscr{NF}}(\Delta)$. One can set $\delta^{(0)}_{\mathscr{NF}}(\Delta) = \Delta$. The $\mathscr{NF}$-number of $\Delta$ is the smallest integer $k > 0$ for which $\delta^{(k)}_{\mathscr{NF}}(\Delta) \simeq \Delta$. In the present paper we are especially interested in the $\mathscr{NF}$-number of a finite graph, which can be regraded as a simplicial complex of dimension one. It is shown that the $\mathscr{NF}$-number of the two copies of complete graph $K_n$ joint by a common vertex on $[2n-1]$ is equal to $2n+2$.

### A Gentle Introduction to Galois Representations Connected with Elliptic Curves

Adrian Kowalski - Politechnika Krakowska im. Tadeusza Kościuszki

In this talk we will cover basics of the extremely deep and rich theory of Galois representations. These are a spectacular application of representation theory to solve deep problems in number theory. We will start by recalling basics of Galois theory and then proceed to see how elliptic curves give rise to Galois representations related to finite Galois extensions in a natural way. Then we will consider representations of the absolute Galois group of rational numbers which is an infinite group with extremely complicated structure. To tame the infinite, we will put a topology on this group and consider continuous representations over topological fields. It will occur that the first idea of examining complex representations is rather flawed and to fix this flaw we will introdue representations over p-adic numbers.

### An application of SAT-solvers to the problem of solving logic puzzles

Maria Książkiewicz - Uniwersytet Kardynała Stefana wyszyńskiego w Warszawie - Polsk

We present basis of working of SAT-Solvers on the example of solving logical and combinatorial riddles. We present some NP-complete problems with their solution: finding a Hamiltonian path in a graph. Then, we present a puzzle about ladies and tigers by Raymond Smullyan.

We present some basic information about classical logics and reductions of riddles to the CNF formula. The results lead us to the conclusion about the way of a representation of tasks in the language of propositional logics and about technique of their transformation into the CNF formuła which enables us to solve the riddles automatically, using a computer.

### Symmetry in Numbers: Concrete Illustration Using Young Diagrams -- PLAKAT

Karol Maciejczyk & Mateusz Mączka - Politechnika Krakowska im. Tadeusza Kościuszki

In our poster, we delve into the captivating realm of Young Diagrams and their significance in mathematics, particularly within the field of algebraic group representation theory. Young Diagrams serve as a unique tool for visualizing integer partitions and are closely tied to sign representations that explore symmetric structures within groups. Our work showcases various integer partitions using these diagrams and explains how they can be correlated with sign representations. We invite you to explore abstract patterns and profound connections between mathematics and symmetry through the straightforward and elegant form of Young Diagrams.

### Big algebras and Dynkin automorphisms

Vladyslav Zveryk - Uniwersytet Jagielloński - Poland

Let $G$ be a complex reductive group and $g$ be its Lie algebra. To any irreducible representation $V$ of $G$ we associate an algebra $\mathrm{End}((V)\otimes S(g))^G$, which we call the Kirillov algebra of this representation. Kirillov algebras encode a variety of properties of representations, for example, commutativity of the Kirillov algebra is equivalent to a representation being weight multiplicity free. We are interested in certain commutative subalgebras of the Kirillov algebra of a representation, namely its center and maximal commutative subalgebras. It appears that the center has a simple description and that there is a maximal commutative subalgebra possessing a rich structure. This algebra, which we call the big algebra, can be constructed using the well-known solution of Vinberg's quantization problem via the Feigin-Frenkel center. During my talk, I will present the construction and properties of big algebras in terms of representation theory and geometry, as well as some further directions in studying them including the so-called Dynkin automorphisms. The talk is based on my research with Tamas Hausel during two summer internships in IST Austria.