In the talk, based on a joint work with R. Auffarth, we will show when the locus of abelian surfaces containing elliptic curves with prescribed exponents is non-empty and when is irreducible. Then, we will make a conjecture of how many components there are in a general case. As an application of the result, we will investigate Jacobians of genus 3 curves that are completely decomposable.

Condensed mathematics is a new theory developed by P. Scholze and D. Clausen, which provides (among many other things) a way of incorporating topological rings into the realm of algebraic geometry. In my talk I will define the crucial notion of condesed abelian groups, which are certain functors on the category of pro-finite sets. I will state basic properties of the category of condesed abelian groups and show how their introdction solves the usual problems one encounters when algebra and topology intertwine.x

This is a continuation of the previous talk. I will introduce solid abelian groups and liquid vector spaces. These notions enable us to embed both non archimedean topological groups and real vector spaces into the very well behaved category of condensed abelian groups. I will also briefly talk about the famous liquid tensor experiment.

I will introduce basic notions and facts on linear algebraic groups, G-spaces and Jordan decomposition.

This talk is based on first four sections of Chapter 2. of the book: T.A. Springer Linear Algebraic Groups

I will continue presentation of basic notions and facts on linear algebraic groups, I will present a proof, based on properties of $G$ spaces that every algebraic group is isomorphic with a closed subgroup of a general linear group $GL$. Finaly, I will prove the Jordan decomposition of an element in linear algebraic group into a product of semi-simple and unipotent elements.

This talk is based on Chapter 2. of the book: T.A. Springer Linear Algebraic Groups

We will continue our presentations on linear algebraic groups. This time we will establish basic results on commutative groups. First we will prove that every commutative group is isomorphic to the product of subgroups of respectively semi-simple and unipotent elements. Then we define the diagonalizable group and show some characterization of such a groups especially the connection between algebraic tori. Finally we will prove classification theorem of connected one dimensional groups which says that every such a group is isomorphic to either multiplicative group $\textbf{G_m}$ or additive group $\textbf{G_a}$.

The talk is based on Chapter 3 of the book T.A.Springer, Linear Algebraic Groups

The classical Recillas' trigonal construction provides a canonical isomorphism between étale double coverings $\tilde C \rightarrow C$ over a trigonal curve $C$ and tetragonal curves $X$ such that the Prym variety of the double covering is isomorphic to the Jacobian of X. There are other instances of n-gonal constructiona giving rise to isomorphisms between some particular abelian varieties. These constructions have important consequences for the understanding of the fibres of the Prym map. We will explain the classical n-gonal constructions and we will focus on the ramified case which extends Donagi’s and Recillas’ results for ramified double coverings. This talk is based on a joint work with Herbert Lange.

T.S.: Multiplier ideals and multiplier events: algebraic geometry at the turn of the millennium I will show a surprising (in 2000) application of multiplier ideals to problems revolving around postulation questions and explain how they possibly seeded a couple of additional grains to the idea of IMPANGA creation

G.K.: A Cascade of Calabi-Yau threefolds I will speak about the paper A cascade of Calabi-Yau threefolds written with Michal Kapustka with an Appendix of Piotr Pragacz.

S.C.: Euler characteristic of a complete intersection I will present a transformation formula for the degree of the Fulton-Johnson class of a complete intersection under a blow-up of a smooth subvariety.

At the beginning of the lecture, I will recall the main properties of non-singular points of an algebraic variety. Then I will switch to defining and studying the Lie algebra of a linear algebraic group.

The talk is based on Sections 4.3 and 4.4 of of the book T.A.Springer, Linear Algebraic Groups

In this talk we will discuss some topological properties of morphisms of general algebraic varietes and apply them to study algebraic groups. In particular, we will prove that a quotient of a linear algebraic group by its closed, normal subgroup exists and is itself a linear algebraic group.

The talk is based on Chapter 5 of the book T.A.Springer, Linear Algebraic Groups

In the talk, we will introduce the basic ingredients of the theory of linear algebraic groups, such as parabolic subgroups, Borel groups and maximal tori. We will show several properties of these subgroups, including the conjugacy theorems for Borel groups and maximal tori. As many results are closely related to solvable groups, we will include the structure theory of connected solvable groups in the presentation.

This talk is based on Chapter 6. of the book: T.A. Springer, Linear Algebraic Groups

In this talk I will discuss special elements of families of double octic Calabi-Yau threefolds. I will briefly remind basic definitions concerning Calabi-Yau threefolds, modularity and Picard-Fuchs operators. I will recall simplest constructions of Calabi-Yau threefolds, then I will discuss double octic Calabi-Yau threefolds.

I will present basic formulations of modularity conjecture, then I will describe basic properties of double octic Calabi-Yau threefolds.

In the third part of my talk I will present the Faltings-Serre-Livné method of proving modularity of two dimensional Galois representations and give some special cases.

I will start with defining representable functors and ind-schemes and stating most important properties od them, mentioning an example od standard grassmannians. Then I will present the definition and first properties of the affine grassmannian of $\operatorname{Gl}_n$.

I will continue the topic of ind-schemes, presenting fundamental results from the theory. Then, I will turn to the affine grassmannian of $\operatorname{Gl}_n$ and study it in detail. In the end, I will briefly state the definition of affine grassmannians associated to general groups and their construction using loop functors.

In my talk will show how we can compute degenerate fiber of a smooth family of double octics by using almost purely combinatorial methods. This is achieved by establishing a few lemmas about resolutions of double covers and by using certain diagrams that track incidences between irreducible components in the branching divisor. I will also briefly mention possible generalizations to higher dimensions.

The talk will be devoted to hyperelliptic curves of genus 3 with two additional (non-hyperelliptic) involutions. I will start by presenting how to use such a rich structure of an automorphism group of a curve to find its defining equation. In addition, we will see that there is a nice way to encode such curves as 2+3 points on the projective line up to projective equivalence. In the second part of the talk, I will focus on Prym varieties of the coverings induced by the involutions with 4 fixed points. In particular, I will describe their period matrices and show that the associated Prym map is a dominant double covering of the moduli space of (1,2)-polarised abelian surfaces containing a complementary pair of elliptic curves of exponent 2.

A covering of curves is called hyperelliptic if both curves are hyperelliptic. It is well known that Prym maps of 2:1 hyperelliptic coverings are never injective. Recently, we have constructed hyperelliptic Klein coverings, i.e. 4:1 hyperelliptic coverings with the Klein four-group of deck transformations. Following a joint paper with Angela Ortega (arXiv:2302.13041) we will show that the Prym maps of such coverings are injective.

A semistable degeneration is a family of algebraic varieties in which generic fibers are smooth and the degenerate fiber admits "nice" singularities. Such families allow us to investigate the cohomological structure of generic fibers by looking at the cohomologies of the degenerate fiber and vice-versa. In my talk I will introduce necessary notions and will present how we can construct such families, starting from a Calabi-Yau threefold. I will also show a few examples of explicit computations of cohomology groups of Calabi-Yau threefolds by using semistable degenerations.

A $p$-adic variety with good reduction has unramified cohomology. The converse is true for elliptic curves and K3 surfaces: if the Galois action is unramified (or crystalline in the $p$-adic setting), then the variety has potentially good reduction. In my talk I will present an example of a Calabi-Yau threefold which shows the failure of this criterion in higher dimension. It is constructed as a resolution of a double cover of the projective space, branched along a union of eight planes. The talk is based on a joint work with Marcin Oczko.

Imagine we want to do a massive road trip to the US and we want to visit all the major cities. Our goal is to find the shortest route that starts in San Francisco, hits every single city exactly once, and then retur to the starting point. This is known as the Travelling salesman problem (TSP), but it has a lot of other formulations. During this lecture we'll talk about the mathematical formulation of this problem, why it is so difficult for computers to solve, and how we might get past those complexities. At the end, we'll see why the TSP is such an important problem and how to apply it in real-life cases.

There are good structure theorems for classifying Gorenstein projective varieties of co-dimension 3 and lower. For instance, co-dimension 3 Gorenstein varieties are given by Pfaffian ideals. The co-dimension 4 and higher case is still an open problem. Unprojection is a technique developed by Miles Reid and Stavros Papadakis for constructing varieties of higher codimension from those of lower co-dimension, that works as a working substitute for a structure theorem for Gorenstein varieties of codimension 4. In this talk I will introduce this idea in generality, followed by several examples of explicit computations. Of special interest are ‘Tom’ and ‘Jerry’ which are two families of Gorenstein co-dimension 4 varieties arising as unprojections.

In this talk I will present a method of computation of all finite subgroups (upto conjugation) of $GL_n(K)$, where $K$ is a quadratic field of class number 1. This methods is based on the Schur bound of the order of such groups and representation theory of finite groups. I will also briefly present our motivation to study this problem.