In this talk I will present definitions and basic properties of Calabi-yau manifolds. I will briefly recall the holonomii group and present geomeatric interpretation of the Hodge numbers.

I will present Tian-Todorov unobstructedness theorem for Calabi-Yau manifolds, I will recall necessary information about deformations of complex manifolds
zoom seminar: https://zoom.us/j/94487954519?pwd=TXZFSy8yY1dTUGo1MSt5VGZpRnMxQT09

I will finish presentation of Tian-Todorov unobstructedness theorem for Calabi-Yau manifolds, I will also present first constructions of Calabi-Yau threefolds as small resolutions of quintic hypersurface in $\mathbb P^4$.
zoom seminar: https://zoom.us/j/95888467582?pwd=bEloNW1YOFdERThucnhEVlJoeDRVQT09

I will present a counter example to Tian-Todorov unobstructedness theorem for Calabi-Yau manifolds in positive characteristic, I will also present basic properties of nodal quintic hypersurfaces in $\mathbb P^4$ (topology of a small resolution, defect, cubic form on the Picard group) and give examples with highest numbers of nodes.
ms teams online seminar

I will discuss small resolutions of nodal hypersurfaces, in particular projectivity criterion and transformation of divisors under smll resolution, I will present nodal. hypersurfaces in $\mathbb P^4$ with large number of nodes
ms teams online seminar.

W tym wykładzie omówię teorię rezolwent wolnych nad pierścieniami Noetherowskimi. Wychodząc od Twierdzenia Hilberta o syzygiach, omówię prace Buchsbauma, Eisenbuda i Hochstera z lat 70-tych i opowiem o ostatnich rezultatach rozwijajacych te teorie.

We consider singular quintic threefolds containing a cone. We show that their minimal resolutions are Calabi-Yau threefolds admitting a type III contraction. We describe the Kähler cone of these varieties. We also construct Calabi-Yau threefolds arising as the smoothing of the manifolds obtained through the type III contraction.
ms teams online seminar.

In this talk I will present basic notions used in the study of the minimal model problem. I will start from Kleiman's criterion for ampleness and then define terminal, canonical and log-terminal singularities.
The talk will be based on: Kawamata, Matsuda and Matsuki Introduction to the Minimal Model Problem, sections 0-1 and 0-2.

We study the behaviour of the Łojasiewicz exponent under hyperplane sections and its relation to the order of tangency. This is a joint work with Christophe Eyral
Zoom online seminar, for link email S. Cynk

My talk is the second one in the series of talks based on the Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki. I will begin with a short statement of the fundamental conjectures of the Minimal Model Program and then I will move on to the topics related to the canonical varieties, including Iitaka dimension, Kodaira's lemma and the notion of a variety of general type. The third part of my talk will focus on the statement and the proof of the Covering Lemma.

In my talk I will discuss several vanishing theorems. I will focus on the Kawamata-Viehweg vanishing theorem and the Elkik-Fujita vanishing theorem and present some of their consequences.
The talk will be based on "Introduction to the Minimal Model Problem" by Kawamata, Matsuda and Matsuki.

In the first part of my talk, I'll focus on the proof of Rationality Theorem. Then I will illustrate how Rationality Theorem can be used to prove Cone Theorem, a structure theorem on the closed cone of curves of an algebraic variety. This theorem is one of the key steps toward the theory of minimal models.
The talk is based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

In the first part of my talk, I'll focus on the proof of the Non-Vanishing Theorem. Then I will illustrate how the Non-Vanishing Theorem can be used to prove the Base Point Free Theorem which in turn will be used to prove the Contraction Theorem.
The talk will be based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

We continue the discussion of the Minimal Model Program with description of types of contractions of extremal rays, namely contractions of fiber type, contractions of divisorial type and contractions of flipping type.
The talk is based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

In this talk we discuss the MMP in the realm of toric varieties. We present Reid's proof that the flip conjecture holds for toric morphisms and describe several examples of flipping contractions and their flips.
The talk is based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

We continue the discussion of the MMP by giving generalizations of Non-Vanishing Theorem and the Base Free Theorem replacing the ample (or nef and big divisors) in the original statements by nef and abundant divisors.
The talk is based on Introduction to the Minimal Model Problem by Kawamata, Matsuda and Matsuki.

I will give a brief introduction to Schubert varieties in generalized Grassmannians, i.e. homogeneous varieties $G/P$, where $P$ is a maximal parabolic subgroup of a simple Lie group $G$. For the minuscule case of $G= E_6$ and $E_7$ I will describe some resolutions of their defining ideals. The motivation comes from the role played by defining ideals of Schubert varieties of low codimension (and their intersections with the opposite big cell) in the structure theory of resolutions of length 3 in Weyman's $T_{p,q,r}$-program.
This is joint work with J. Torres and J. Weyman.

This will be the presentation of my bachelor thesis. The aim of this seminar will be to present proof that the j-invariant of an elliptic curve with complex multiplication is an algebraic integer. I will begin with preliminary information including basic facts about imaginary quadratic fields and elliptic curves with complex multiplication and then I will move on to the topics related to the modular forms and modular polynomials. The third part of my talk will focus on the proof of the main theorem. I will conclude with a nice application of this theory to justify why Ramanujan Constant is almost integer.

I will give a gentle introduction to Littelmann paths and how they encode bases of irreducible representations of semi-simple Lie algebras. Amongst the advantages of this technique are branching rules which describe the decomposition into irreducible components of the restriction of a given irreducible representation to a Levi subalgebra. I will present a similar branching rule, for a non-Levi case that seems to be quite special. This is joint work with Bea Schumann.

Good semigroups are defined as a family of submonoids of $N^d$, properly containing the family of value semigroups of rings of curves singularities. In the case $n=1$, they are just numerical semigroups and their properties, in relation with those of the associated monomial curves have been well studied. Extending those results to the general case, we construct and study the Apéry sets of good semigroups and, focusing mostly on the case $n=2$, we show how these sets are used to describe properties such as symmetry and almost-symmetry. In the case of value semigroups of curves, such properties correspond to Gorensteiness and almost-Gorensteiness of the associated rings.

We give a formula for pushing forward the classes of Hall-Littlewood polynomials in Grassmann bundles. This formula generalizes Gysin formulas for Schur $S$-functions. Moreover it establishes a formula for Schur $P$-functions with the help of Gaussian polynomials.

Analogously to the work which followed the classification of finite symplectic actions on K3 surfaces by S. Mukai in 1988, we will use the paper Finite groups of symplectic automorphisms of hyperkähler manifolds of type $K3^{[2]}$ by G. Höhn and G. Mason to study some very symmetric $K3^{[2]}$ type fourfolds, i.e ones acted upon via (not necessarily symplectic) automorphisms by the largest possible groups strictly containing the maximal groups from the article.

We construct Calabi-Yau manifolds of arbitrary dimensions as a resolution of a quotient of a product of a K3 surface and (n-2) elliptic curves with a strictly non-symplectic automorpism of order 2, 3, 4 or 6. This construction generalize a result of Cynk and Hulek and the classical construction of Borcea and Voisin, the proof is based on toric resolution of singularities. Using Chen-Ruan orbifold cohomology and Yasuda theorem we compute the Hodge numbers of all constructed examples. Moreover, using Rosen’s result we give a method to compute the local Zeta functions.

Let $\mathcal{A}=(a_n)_{n\in\mathbb{N}_+}$ be a sequence of positive integers. The function $p_\mathcal{A}(n,k)$ counts the number of multi-color partitions of $n$ into parts in $\{a_1,\ldots,a_k\}$. We examine several arithmetic properties of the sequence $(p_\mathcal{A}(n,k) \pmod{m})_{n\in\mathbb{N}}$ for an arbitrary fixed integer $m\geqslant2$, and apply them to the special cases of $\mathcal{A}$. In particular, for a fixed parameter $k$, we investigate both the upper bound for the odd density of $p_\mathcal{A}(n,k)$ and the lower bound for the density of $\{n\in\mathbb{N}: p_\mathcal{A}(n,k)\not\equiv0 \pmod{m}\}$. Furthermore, we present some new results related to restricted $m$-ary partitions and state a few open problems at the end of the talk.

Finding equations of elliptic curves with complex multiplication poses a surprisingly difficult task. During my presentation I will introduce some necessary theory to tackle this problem. I will also describe an algorithm developed by H.M. Stark, that enables us to find equations of such curves and will show some examples of explicit formulas for different curves.

We present an algorithm for counting points on a double octic Calabi-Yau threefold associated to a configuration of eight planes over a finite field based on the existence of an elliptic curve fibration.

This paper contains results of PhD studies interrupted by sudden death of Aleksander Czarnecki on February 28, 201.

aleksander.czarnecki.im.uj.edu.pl

Given a one-dimensional family of Calabi-Yau threefolds, the variation of its third cohomology defines a local system on the parametrizing space. This local system is isomorphic to a one coming from the certain differential operator, called the Picard-Fuchs operator of the given family. In my talk I will introduce basic properties of local systems associated with Picard-Fuchs operators; I will also introduce monodromy representations as the basic mean of studying them. In particular, results concerning the transition matrix between local Frobenius bases at different singular points will be presented. The talk is based on my Master's Thesis.

Given a set $X$ consisting of subspaces of $\mathbb{P}^n$ one can ask whether vanishing along $X$ imposes independent conditions on forms of degree $t$. If this is the case, we say that $X$ has good postulation in degree $t$. By the well-known Hartshorne-Hirshowitz Theorem, a set of general lines in $\mathbb{P}^3$ has good postulation in any degree. That result leads to the study of the postulation of general planes in $\mathbb{P}^4$. In my talk I will present the method to show that the set of general planes has good postulation in certain degrees.

In 2005, M. Dumnicki and W. Jarnicki presented an algorithm which can be used to find the upper bound of the dimension of linear system of plane curves of given degree, with multiple points in general position. This algorithm, called the reduction method, can be generalized to the n-dimensional case. In my talk I will present this method in the 2-dimensional case, along with the detailed proof why it works. Then I will present a generalized, $n$-dimensional version of the reduction method, along with the general proof and examples.