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Algebra Seminar at UJ

The Algebra Seminar at UJ will be taking place on Thursday morning at 10am (Europe/Warsaw), unless otherwise stated. This term the seminar is online over MSTeams. If you are interested in receiving announcements about upcoming seminars, please email guelor@guelan.com. 

• Past seminars

Summer semester 2026

Winter semester 2026

Summer semester 2025

TBA

Winter semester 2025

TBA

Summer semester 2024

TBA

Winter semester 2024

TBA

Summer semester 2023

TBA

Winter semester 2023

Abstracts

Cedric Bonnafe - Equivariant cohomology of smooth Calogero-Moser spaces

oint work with Peng Shan. We explain how the representation theory of rational Cherednik algebras at t=0 can be used to solve a conjecture of Ginzburg and Kaledin on the cohomology of smooth Poisson deformations (or of symplectic resolutions) of symplectic quotient singularities. We deduce an intriguing property of the character table of the symmetric group.

Pietro Campochiaro - Kunz cone and Apéry resolution of numerical semigroups

Numerical semigroups are submonoids of the set of non-negative integers equipped with the usual sum operation that have finite complement with N. The smallest positive element of a numerical semigroup is called multiplicity. The numerical semigroups with fixed multiplicity greater than one are parametrized by the integral points of a pointed cone, known as the Kunz cone. If two integral points lie in the interior of the same face of the Kunz cone, their corresponding numerical semigroups share many properties, for instance their defining toric ideals have the same Betti numbers. In the first part of this talk I will explain the definitions and some basic properties of numerical semigroups and the Kunz cone, and then, following arXiv:2310.03612v2, describe the Apéry resolution of a numerical semigroup and show some of its applications.

Viola Conte - Introduction to cluster algebras (second part)

This talk is a continuation of the previous one. I will focus on the algebraic and representation-theoretic counterparts of quiver mutation and on their generalization to weighted quivers. I will present mutations of path algebras and species, especially those arising from triangulated orbifold surfaces. Part of this material is related to my PhD project.

Viola Conte - Introduction to cluster algebras (first part)

In this talk, I will introduce the basic definitions of cluster algebras and of (weighted) quiver mutation through some examples. In particular, I will discuss their relations with triangulated orbifold surfaces, and explain how this connection leads to the classification of cluster algebras of finite mutation type.

Prajwal Samal - Construction of Fourier Mukai Functors Using GLSM

Gauged Linear Sigma Model (GLSM) is a relatively novel, physics inspired technique that utilizes the geometry of variation of GIT to construct functors between derived categories of varieties appearing as critical loci in the GIT quotients. In many cases, these functors turn out to be fully-faithful or even equivalences. This is done by relating the derived categories of interest to matrix factorization categories of certain Landau-Ginsberg models. In this talk, I will discuss this circle of ideas using examples. I will also present a recent result where we used this framework to construct a derived equivalence between two non-birational Calabi-Yau threefolds.

Tymoteusz Chmiel - Semi-stable models and Galois representations

I will start by describing Galois representations arising from the cohomology of an algebraic variety. Then I will introduce the notion of a semi-stable model of such a variety, and explain how one can use it to compute the associated Galois representation. As an application, I will describe an example of a Calabi-Yau threefold with bad reduction modulo prime p>3 but unramified cohomology. The talk is based on a joint work with Marcin Oczko.

Jakub Chmiel - Characterization of Growth Functions of Algebras, Semigroups, and Languages

We will discuss a theorem of Bell and Zelmanov on the possible growth functions of finitely generated algebras, finitely generated semigroups, and hereditary languages. The theorem shows that, up to asymptotic equivalence, these three settings give exactly the same growth types. It also gives a precise characterisation of the functions that can occur, formulated in terms of their discrete derivatives. We will sketch the main ideas behind the construction of algebras realising prescribed growth. 

Rafał Mach - Equivalences of derived categories and K3 surfaces

We consider derived categories of coherent sheaves on smooth projective varieties. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces. The talk will be based on "Equivalences of derived categories and K3 surfaces" by Dmitri Orlov.

Calla Tschanz - From logarithmic Hilbert schemes to degenerations of hyperkähler varieties

In this talk, I will discuss my previous work on constructing explicit models of logarithmic Hilbert schemes. This relates to work or Li-Wu on expanded degenerations, Gulbrandsen-Halle-Hulek on degenerations of Hilbert schemes of points and Maulik-Ranganathan on logarithmic Hilbert schemes. The constructions I consider are local. I will then explain how we globalise these in joint work with Shafi and apply them to construct minimal type III degenerations of hyperkähler varieties, namely Hilbert schemes of points on K3 surfaces.

Haleh Hamdi - A characterization of t-Almost Dedekind Domains via w-Secondary Modules

As a dual notion to the primary decomposition of modules over a commutative ring, the concept of secondary representation has been studied since 1973 by Macdonald. A nonzero unitary module M over a commutative ring R with identity is said to be secondary if, for each r in R, either rM = M or r^nM=0 for some positive integer n. A secondary representation of M is an expression M = N_1 + ... + N_t, where each N_i is a secondary submodule of M for i = 1,..., t. If the sum of the N_i is direct, then M is said to be strongly representable. An integral domain R is almost Dedekind, that is, locally discrete valuation ring, if and only if R is one-dimensional and every representable R-module is strongly representable. Thanks to the existence of the generalizations of almost Dedekind domains with respect to the w-operation, we will focus on the w-operation analogue of the aforementioned result by defining w-secondary modules, w-representable modules and strongly w-representable modules. This talk is based on a joint work with H. Kim and H. Baek [1] and supported by FCT2023.06156.CEECIND, and FCT 10.54499/UID/04621/2025 of CEMAT, University of Lisbon.

Andrzej Sitarz - Geometry of algebras

I'll review the program of using noncommutative geometry to extend the geometrical notions using algebra and its tools - and illustrate them by simple examples. I'll mention the applications of this program to physics.

Nikon Kurnosov - Deformations Lagrangian subvarieties of holomorphic symplectic manifolds

In this talk I will highlight an approach to study deformations of holomorphically symplectic manifolds which preserve Lagrangian subvarieties. In the case of Kahler manifold and smooth subvarieties the problem was solved by Voisin, later C. Lehn extended her approach to the snc subvarieties. We approach this question from scratch and focus on understanding the deformations of C-symplectic structures, which work for large classes of Kahler and non-Kahler manifolds, and the Lagrangian subvarieties of a Kummer-type. Joint with M. Verbitsky.

Piotr Pokora - On homological types of hypersurfaces

We introduce the notion of the type of a reduced complex plane curve. This invariant encodes homological properties of the Milnor algebra associated with the Jacobian ideal of the curve. We show that curves of type 0 are precisely the free curves, while curves of type 1 are exactly the plus-one generated curves. After establishing the basic properties of the type, we discuss its conjectural generalization to surfaces in P^3. In contrast to the case of plane curves, we exhibit a family of hypersurfaces for which the type can be arbitrarily small.

Jakub Byszewski - Periodic points of endomorphisms of algebraic groups

Consider an endomorphism of an algebraic group over an algebraically closed field of positive characteristic. We are interested in counting the number of its periodic points of period n. In many classical situations (such as when the endomorphism is the Frobenius map or when the algebraic group is semisimple), this number is given by a nice formula that is cohomological in nature and satisfies a linear recurrence in n; this fact can be equivalently expressed in terms of the rationality of the associated Artin--Mazur dynamical zeta function. In general, however, the formula for the number of periodic points of period n depends also on the p-adic properties of n. We will discuss the general case including the questions of rationality of the associated zeta function and the analogues of the prime number theorem. We will also discuss several examples and relations with cellular automata and topological dynamics. The talk is based on joint work with Gunther Cornelissen and Marc Houben.

Bárbara Muniz - Structure and geometry of the tableau algebra (second part)

We will study the variety associated to the tableau algebra. We classify its maximal ideals, describe the topology of its maximal spectrum and construct a toric embedding. This variety appears, via work of Gonciulea–Lakshmibai, as a toric degeneration of certain partial flag varieties. As an application, we enumerate the corresponding Plücker relations.

Jacinta Torres - Structure and geometry of the tableau algebra (first part)

Motivated by the Minkowski sum of Gelfand-Tsetlin patterns, I will introduce an algebra generated by semi-standard Young tableau with a very simple operation- row concatenation. Then I will present some of its algebraic and geometric properties.

Divya Setia - Tensor product of Demazure modules and Demazure crystals

Given a finite-dimensional simple Lie algebra g, its associated current Lie algebra is denoted by g[t] that consists of the polynomial maps from C to g. In this talk, we explore the structure of tensor product modules for sl2[t], focusing on Demazure modules, which are representations of a Borel subalgebra of an affine Kac-Moody Lie algebra. A Demazure module is said to be of level l if the central element of the affine Kac-Moody Lie algebra acts on the highest weight representation by a scalar l. We are mainly interested in the tensor product of g-stable Demazure modules. Our results are inspired by the question asked by Joseph Anthony that whether the tensor product of Demazure modules admits a Demazure flag, which are filtrations by sub-modules whose successive quotients are Demazure modules. We prove that the tensor product of two level 1 Demazure modules has a Demazure flag of level 2 for current Lie algebra sl2[t]. A similar problem can also be explored at the level of crystals. For a Demazure module, we have a corresponding Demazure crystal, and it is not necessary that the tensor product of two Demazure crystals can be decomposed as a disjoint union of Demazure crystals. At the end, I will also explain the necessary and sufficient condition of the decomposition of two Demazure crystals as a disjoint union of Demazure crystals for a symmetrizable Kac-Moody Lie algebra.

Marco Rampazzo - Semiorthogonal decompositions for Fano varieties

Semiorthogonal decompositions provide a way to describe a triangulated category in terms of a collection of simpler components. In this context, the structure of the derived categories of coherent sheaves on Fano varieties is the subject of several conjectures. In this talk, I will present some examples and discuss related open problems.

Pietro Campochiaro - Some classes of one-dimensional rings characterized by their reflexive ideals

Reflexivity of modules and ideals and its relation with the properties of the ring is a classical topic of study. A fractional ideal I of a ring R is reflexive if it coincides with its divisorial closure, that is, if I=R:(R:I). I will discuss some basic properties of regular reflexive fractional ideals of one-dimensional Cohen-Macaulay local rings, and how these ideals can be used to characterize almost-Gorenstein rings, rings of minimal multiplicity, and Arf rings. Finally, I will explain how numerical semigroups constitute a useful tool for studying reflexive ideals of rings. This talk is based on a joint work with Marco D’Anna and Francesco Strazzanti.

Lyalya Guseva - Full Exceptional Collections on Isotropic Grassmannians

The bounded derived category of coherent sheaves, D(X), is an important invariant of an algebraic variety X. While the structure of derived categories is generally quite intricate, in certain cases, when D(X) admits a so-called full exceptional collection, it can be described explicitly. Some of the earliest examples of full exceptional collections were constructed by Kapranov in 1983 for classical Grassmannians. Since then, a well-known conjecture has suggested that full exceptional collections consisting of vector bundles exist in the derived categories of all rational homogeneous varieties. In my talk, I will outline the proof of this conjecture for all rational homogeneous varieties associated with symplectic groups. This is joint work with Sasha Novikov.

Benedetta Piroddi - Automorphisms on relative Prym varieties induced by K3 surfacess

The relative Prym construction is a way to produce examples of symplectic varieties, starting from a K3 surface S that is a double cover of a del Pezzo (or Enriques) surface T. For some special choices of a curve D on S, the moduli space M of sheaves on S with Mukai vector v = (0, |D|, 1 - g(D)) is singular, and it has a non-natural, non-symplectic regular involution that acts by taking the duals of stable sheaves. The composition of this involution with the one induced on M by the covering involution ι of S → T is symplectic, and the biggest component of its fixed locus, the relative Prym variety, is a symplectic variety. In this talk, I'm going to assume that a bigger group G acts on S, such that the action of G/ι is symplectic; I'll discuss the conditions under which this latter action extends to a symplectic action on the relative Prym variety, and give some examples. This is a joint work in progress with Annalisa Grossi and Sasha Viktorova.

Tomasz Wawak - Very symmetric hyper-Kähler manifolds

TBA

Vladyslav Zveryk - Kirillov Algebras I

The talk will be the first in the series of upcoming talks of mine aimed to present my research at IST Austria. I will start with a short overview of the whole project, followed by an introduction to the theory of vertex algebras. The main example will be the vertex algebra associated to an affine Kac-Moody algebra, which will play a crucial role in the future talks.

Lorenzo Guerrieri - Gorenstein licci ideals of deviation 2

I will describe some old results towards the classification of licci Gorenstein ideals of deviation 2, following the PhD thesis of Elias Lopez.

Vladyslav Zveryk - Kirillov Algebras II

The talk will be the second in the series of talks aimed at presenting my research at IST Austria. I will give an introduction to the theory of vertex algebras, starting with their definition and then deriving their main properties. Attendance at the previous talk is not required.

Tymoteusz Chmiel - TBA

TBA

Maciej Dołęga - Rationally weighted b-Hurwitz numbers via W-algebras

Weighted Hurwitz numbers were introduced by Harnad and Guay-Paquet as an object covering a wide class of Hurwitz numbers of various types. A particularly strong property of Hurwitz numbers is that they are widely governed by topological recursion (TR) of Chekhov--Eynard--Orantin. The program of understanding how TR can be used to compute various Hurwitz numbers was carried over the last two decades by considering each case separately, and finally the general case of rationally-weighted Hurwitz numbers was proved recently by Bychkov--Dunin-Barkowski--Kazarian--Shadrin. We are going to discuss a more general case of weighted $b$-Hurwitz numbers. We show that their generating function can be associated with the Whittaker vectorfor a W-algebra of type A. In particular it satisfies W-constraints that are the Airy structure -- an algebraic reformulation of the concept of TR due to Kontsevich--Soibelman. Our result gives a new explanation of remarkable enumerative properties of Hurwitz numbers and extends it to the $b$-deformed case. This is a joint work with Nitin Chidambaram and Kento Osuga.

Jakub Koncki - Nakajima's creation operators and the Kirwan map

The Hilbert scheme of points in the affine complex plane is a smooth variety. Several descriptions of its cohomology groups are known. One may use Białynicki-Birula decomposition, Nakajima creation operators, or the Kirwan map. I will present a relation between the last two of the mentioned methods. I will describe the action of Nakajima's creation operators on the characteristic classes of the tautological bundle. This is joint work with Magdalena Zielenkiewicz.

Anna Szumowicz - Bounding Harish-Chandra characters

Let $G$ be a connected reductive algebraic group over a $p$-adic local field $F$. We study the asymptotic behaviour of the trace characters $\theta _{\pi}$ evaluated at a regular element of $G(F)$ as $\pi$ varies among supercuspidal representations of $G(F)$. I give the sketch of the proof that for $G$ semisimple the trace character is uniformly bounded on $\gamma$ under the assumption, which is believed to hold in general, that all irreducible supercuspidal representations of $G(F)$ are compactly induced from an open compact modulo center subgroup of $G(F)$. If time allows I could also discuss progress in finding explicit bounds on the trace characters.

Oana Veliche - On the minimal free resolution of the residue field

In a paper from 1968, Golod proved that the Betti sequence of the residue field of a local ring attains the upper bound given by Serre if and only if the homology algebra of the Koszul complex of the ring has trivial multiplications and trivial Massey operations. This is the origin of the notion of Golod ring. Using the Koszul complex components as building blocks Golod also constructed a minimal free resolution of the residue field of a Golod ring. With Van Nguyen, we extend this construction for an arbitrary local ring, up to homological degree five, and explicitly show how the multiplicative structure of the homology of the Koszul algebra is involved, including the triple Massey products. The talk will illustrate this construction and various consequences of it.

Pedro Macias Marques - Reducible families of Artinian Gorenstein algebras

We study local Artinian Gorenstein (AG) algebras and consider the set of Jordan types of elements of the maximal ideal, i.e. the partition giving the Jordan blocks of the respective multiplication map. In joint work with Tony Iarrobino, we construct examples of families Gor(T) of local AG algebras with given Hilbert function T, and use obstructions that the symmetric decomposition of the associated graded algebra of an AG algebra A imposes on the Jordan type of A to study their irreducible components.

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