The goal of this talk is to present definition and basic properties of periods of a modular form for a congruence subgroup $\Gamma \subset \operatorname{SL}_2(\mathbb Z)$. I will start with a brief introduction of modular sumbols.

I will briefly recall definition of modular forms and the associated $L$-functions. Then I will introduce the period polynomials and present their basic properties.

Let $X \longrightarrow C$ be a projective family of surfaces over a curve with smooth general fibres and simple normal crossing singularity in the special fibre $X_0$. The motivating example for the type of degeneration considered is a type III good degeneration of K3 surfaces. We construct a good compactification of the moduli space of relative length $n$ zero-dimensional subschemes on $X\setminus X_0$ over $C\setminus\{0\}$. In order to produce this compactification we study expansions of the special fibre $X_0$ together with various GIT stability conditions, generalising the work of Gulbrandsen-Halle-Hulek who use GIT to offer an alternative approach to the work of Li-Wu for Hilbert schemes of points on simple degenerations. We construct stacks which we prove to be equivalent to the underlying stack of some choices of logarithmic Hilbert schemes produced by Maulik-Ranganathan. The stacks we construct offer models of good type III degenerations of Hilbert schemes of points on K3 surfaces.

I will briefly recall classical Hodge theory including:

• De Rham cohomology, harmonic forms, the Hodge isomorphism theorem
• Kähler manifold, the Hodge decoposition theorem
• Hodge structure of weight k on a $\mathbb Z$-module, Hodge structure on homology and cohomology.

This talk is based on section 1.1 of the book: C. Peters, J.Steenbrink, Mixed Hodge Structures

This is the second part of the talk in which I am recalling the following topics

De Rham cohomology, harmonic forms, the Hodge isomorphism theorem

Kähler manifold, the Hodge decoposition theorem

Hodge structure of weight k on a $\mathbb Z$-module, Hodge structure on homology and cohomology. This talk is based on section 1.1 of the book: C. Peters, J.Steenbrink, Mixed Hodge Structures

In this talk, I will cover Lefschetz Decomposition. I will present the following sub-topics and discuss some basic theorems in the end concerning these topics

• Representation Theory of SL(2, R)
• Primitive Cohomology

The exposition will be mainly based on section 1.2 of the book: C. Peters, J.Steenbrink, Mixed Hodge Structures.

In this talk, I will cover Lefschetz Decomposition. I will present the following sub-topics and discuss some basic theorems in the end concerning these topics

• Representation Theory of SL(2, R)
• Primitive Cohomology

The exposition will be mainly based on section 1.2 of the book: C. Peters, J.Steenbrink, Mixed Hodge Structures.

In this talk we discuss the Galois representations of abelian varieties. In particular, we focus on the "exponential'' (i.e. $l$-power) torsion of the Jacobian of the curve $y^l = f(x)$. We determine the image of the associated l-adic representation up to the determinant under some mild conditions on the polynomial $f(x)$. We show also that the image of the determinant is contained in an explicit $\mathbb Z_l$-lattice with finite index. As an application we prove the Mumford--Tate conjecture for a generic superelliptic Jacobian.

I will start by recalling the standard constructions related to Hodge structures. I will then switch to defining and stating the main properties of the Mumford-Tate groups on Hodge structures.

The talk will be based on Sections 2.1 and 2.2 of the book Mixed Hodge Structures by C. Peters and J. Steenbrink.

I will start by recalling the standard constructions related to Hodge structures. I will then switch to defining and stating the main properties of the Mumford-Tate groups on Hodge structures.

The talk will be based on Sections 2.1 and 2.2 of the book Mixed Hodge Structures by C. Peters and J. Steenbrink.

We’ll start by defining the Hodge to de Rham spectral sequence describing the relation between Dolbeault and de Rham cohomology on complex manifold and show that for compact Kähler manifolds, the sequence degenerates at $E_1$ page. This result, together with a certain fact regarding Bott-Chern cohomology will be used to define a Hodge decomposition in a strong sense. We'll show that under certain assumptions regarding a so-called "putative Hodge filtration" on $H^k(X, \mathbb C)$ and $H^{2n-k}(X, \mathbb C)$, a Hodge decomposition in a strong sense can be recovered. Finally, the notion of Hodge complex – and, in particular, Hodge complex of sheaves – will be discussed.

The talk will be based on Section 2.3 of the book Mixed Hodge Structures by C. Peters and J. Steenbrink.

We’ll start by defining the Hodge to de Rham spectral sequence describing the relation between Dolbeault and de Rham cohomology on complex manifold and show that for compact Kähler manifolds, the sequence degenerates at $E_1$ page. This result, together with a certain fact regarding Bott-Chern cohomology will be used to define a Hodge decomposition in a strong sense. We'll show that under certain assumptions regarding a so-called "putative Hodge filtration" on $H^k(X, \mathbb C)$ and $H^{2n-k}(X, \mathbb C)$, a Hodge decomposition in a strong sense can be recovered. Finally, the notion of Hodge complex – and, in particular, Hodge complex of sheaves – will be discussed.

The talk will be based on Section 2.3 of the book Mixed Hodge Structures by C. Peters and J. Steenbrink.

Studiowanie parametrycznych rodzin krzywych eliptycznych niesie ze sobą pokusę znalezienia formuły dla sumy potęg odchyleń od liczby punktów na prostej rzutowej dla każdej krzywej z rodziny, zwanej k-tym momentem rodziny. Każdy moment niesie inną informację o zadanej rodzinie. Dla rodzin jednoparametrowych opowiem o tym, jak policzyć momenty w pewnych szczególnych przypadkach. Sytuacja dla pierwszego momentu jest całkiem dobrze poznana, dla drugiego we wspólnej pracy z Matiją Kazalickim udowadniamy związek drugiego momentu dla rodziny kubik z pewnymi krzywymi wyższych genusów. Podobnie jak dla pierwszego momentu, drugi posiada pewne naturalne ,,skrzywienie'' w kierunku ujemnym. Korzystając z hipotezy Sato-Tate'a dla krzywej genusu 2, udowodnimy, jak pokazać istnienie tego skrzywienia. Wspomnę również o zastosowaniu metody momentów do dowodzenia bez metody Faltingsa-Serra modularności pewnych rozmaitości Calabiego-Yau.

After a brief introduction into theory of complex tori and abelian varieties we will focus on the notion of a dual abelian variety. We will show examples of non-principally polarised abelian varieties of any type that are not isomorphic to their duals. Moreover, we will show an example of an abelian variety that is isomorphic to its dual, yet it is not principally polarisable.

The talk is based on a joint work with Aleksandra.

We know thanks to the work of L. Giovenzana, A. Grossi, C. Onorati and D. Veniani that OG10-type hyperkähler manifolds do not admit any non-trivial finite symplectic automorphisms. What about non-regular symplectic birational transformations? Given a cubic fourfold V, one can construct a hyperkähler manifold XV of OG10-type following a construction of R. Laza, G. Saccà, C. Voisin. Such manifolds are known as LSV manifolds. It can be shown that any symplectic automorphism on V induces a symplectic birational transformation on XV. In a couple of works with L. Marquand, we study and classify all possible cohomological actions on the OG10-lattice which can be realised as symplectic birational transformations. By investigating further the induced action on cohomology, we exhibit a criterium to decide which of these actions can be realised as induced from a cubic fourfold on an associated LSV manifold.

In the first part I will describe different ways of associating cohomology classes (in different cohomology theories) to subvarieties of compact algebraic manifolds. In particular, I will discuss fundamental Hodge classes and Thom classes. In the second part I will talk about V-manifolds, which are locally quotients of manifolds by the action of a finite group. The main result in this part of the talk will be the existence of a Hodge structure on the rational cohomology of any amost Kähler $V$-manifold.

The talk will be based on sections 2.4 and 2.5 of Mixed Hodge Structures by C. Peters and J. Steenbrink.

At the beginning of the talk we will define the category of $R$-mixed Hodge structures, provide some examples and show that this category is abelian. Then we will discuss three important filtrations related to a complex $K$ in an abelian category with an increasing filtration $W$ and a decreasing filtration $F$; the three filtrations are on the spectral sequence induced by $F$. Finally we will define and discuss mixed Hodge complexes, in particular we will show how one could construct a mixed Hodge structure from a mixed Hodge complex of sheaves.

The talk will be based on sections 3.1, 3.2 and 3.3 of Mixed Hodge Structures by C. Peters and J. Steenbrink.

We will discuss several familiar notions in the context of mixed Hodge structures. First, we’ll introduce the Mixed Cone and show that it has the properties of the mapping cone. Then, we will study extensions in the category of mixed Hodge structures. In particular, we will cover the group of mixed Hodge extensions and its relation to the Jacobian. At the end, we will briefly review the absolute Hodge cohomology.

The talk will be based on sections 3.4 and 3.5 of Mixed Hodge Structures by C. Peters and J. Steenbrink.

In the talk, we will present recent applications of enumerative algebra to the study of stationary states in physics. Our point of departure are classical Newtonian differential equations with nonlinear potential. It turns out that the study of their stationary states is closely related to solving systems of special polynomial equations. As observed by physicists, the number of solutions is smaller than predicted by the Bezout bound or its celebrated refinement due to Bernstein and Kushnirenko. We will present a generalization of these two bounds based on the theory of Khovanskii bases. Our results give the correct, expected numbers for the polynomial systems that inspired us to look at the more general theory.

The Prym map $P_{g,r}$ assigns to a degree $2$ morphism $\pi : D \longrightarrow C$ of smooth complex irreducible curves ramified in an even number of points $r$, a polarized abelian variety $P(D,C)$ of dimension $g − 1 + r/2$ , where g is the genus of $C$. The variety $P(D,C)$ is called the Prym variety of the covering. The case $r=0$ is very classical and have been studied since the second half of the 19th century. More recently, the ramified case has been considered, and several theorems about Generic and Global Torelli theorems have been obtained. In this talk we will report on a joint work with I. Spelta and P. Frediani concerning the geometry of the fibers of the ramified Prym map for low genus. More precisely, we describe the geometry of the generic fibers of $P_{g,r}$ for $r=2, 1\le g \le 4$, and $r=4, g=1,2$, which are the only cases where the generic fibers o are positive dimensional.

We show that cohomology groups of smooth complex varieties admit mixed Hodge structures. To do this, we introduce the notion of logarithmic de Rham complex and residue maps to show that cohomology groups can be computed via (hypercohomologies of) this log-complex, on which we then put the structure of mixed Hodge complex (of sheaves) from which we recover the desired mixed Hodge structure on the groups themselves.

The talk will be based upon sections 4.1-4.3 of Mixed Hodge Strctures by C. Peters and J. Steenbrink and Théorie de Hodge II by P. Deligne.

We show that cohomology groups of smooth complex varieties admit mixed Hodge structures. To do this, we introduce the notion of logarithmic de Rham complex and residue maps to show that cohomology groups can be computed via (hypercohomologies of) this log-complex, on which we then put the structure of mixed Hodge complex (of sheaves) from which we recover the desired mixed Hodge structure on the groups themselves.

The talk will be based upon sections 4.1-4.3 of Mixed Hodge Strctures by C. Peters and J. Steenbrink and Théorie de Hodge II by P. Deligne.

We will define logarithmic structures on algebraic varieties. We will also prove that mixed hodge structure on a smooth algebraic variety are independent of compactification. In the end we will show some applications of mixed hodge structures to Lefschetz pencils.

The talk will be based on sections 4.4 and 4.5 of the book Mixed Hodge Strctures by C. Peters and J. Steenbrink and Théorie de Hodge II by P. Deligne..

The goal of the talk is to present some results on the automorphisms of generalized Gaudin algebras induced from Dynkin automorphisms of simple Lie algebras. I will start by giving an overview of the construction of generalized Gaudin algebras via the Feigin-Frenkel center, as well as a recollection of the basic properties of Dynkin automorphisms of simple Lie groups. Using the geometry of opers whose rings of functions are isomorphic to various generalized Gaudin algebras, we will link generalized Gaudin algebras associated with a simple Lie algebra and its subalgebra fixed by a Dynkin automorphism. One application of this linkage is a new proof of the global version of Jantzen's twining formula. The talk will be based on arxiv:2311.11872.

In the talk we will present the results concerning the finite group actions on hyper-Kähler manifolds of K3^[n] type. In the first part of the talk, we will explain the strategy of classification of such actions, then in the second half, we will construct examples of manifolds with finite group actions of large cardinality as double EPW sextics and double EPW cubes. Based on the planned PhD thesis.

We will define logarithmic structures on algebraic varieties. We will also prove that mixed hodge structure on a smooth algebraic variety are independent of compactification. In the end we will show some applications of mixed hodge structures to Lefschetz pencils.

The talk will be based on sections 4.4 and 4.5 of the book Mixed Hodge Strctures by C. Peters and J. Steenbrink and Théorie de Hodge II by P. Deligne..

With every linear code one can associate a lattice, and with a lattice - a theta function. It is then, perhaps, not surprising that there is a connection (a homomorphism!) between suitable spaces of codes and modular forms. What is remarkable, is the level of this resemblance: in both worlds there are functions invariant under an action of a group, notions of cusp forms and Hecke operators, also projections and lifts between different geni. The compatibility between these objects will be explained in the first part of the talk. This will prepare the ground for the second part: an adaptation of the doubling method in the framework of self-dual linear codes. The new phenomena that will be presented are the effect of joint work with Thanasis Bouganis.