Classical Prym varieties are principally polarised abelian varieties arising from algebraic curves and they can be seen as a generalisation of the Jacobian of a curve. More precisely, to every double covering between smooth curves we can associate in a natural way a polarised abelian subvariety of the Jacobian of the covering curve. This construction induces a map between the moduli space of double coverings of curves and the moduli space of principally polarised abelian varieties, known as the Prym map. In this talk we will explain the rich and beautiful geometry displayed in the fibres of the Prym map for coverings over low genera curves. We will also present the known results for the Prym map corresponding to coverings between curves of higher degree, which lead to non-principally polarised abelian varieties .

A key result in Wiles' proof of Fermat's last theorem was proving part of a conjecture of Shimura and Taniyama regarding modularity of elliptic curves. The main purpose of the present talk is to explain (to a general audience) the notion of modularity, together with the state of the art on modularity results and its importance on the study of arithmetic properties of varieties. I will conclude the talk explaining how modularity is used while studying solutions of Diophantine equations.

The theory of integral closure of ideals, originating in the early twentieth century with work of Krull, Zariski, Rees, and others, remains a vibrant area of research in commutative algebra. This theory's significance stems from its connections with valuations, which enable a wide range of applications in other mathematical fields such as combinatorics, algebraic geometry, and number theory. Asymptotic properties of integral closures can be understood through multiplicities, which have their origins in intersection theory, developed by Hilbert, Noether, Van der Waerden, and others. In the 1950s, advancements by Samuel, Serre, and Rees brought multiplicities to the forefront of commutative algebra, leading to extensive further development in various directions, including recent works. This talk will provide an overview of the history and key results in these areas, highlighting some open problems along the way.

Symmetries of classical objects (such as Archimedean solids and smooth manifolds?) are described mathematically by groups and their actions. Groups are often inadequate for describing quantum objects such as those that emerge naturally in many fields of mathematics and physics, such as von Neumann algebras, quantum groups, low-dimensional topology, vertex operator algebras, statistical mechanics and condensed matter physics.

The mathematical structure that unifies these different settings are tensor categories; these are interpreted as quantum symmetries.

In this talk, we will start by introducing some of the basic definitions and properties of fusion, braided, and modular categories. We will give some concrete examples to get a better understanding of their structures.

We will also present the main invariants and properties of modular categories and give an overview of the current situation of the classification program for modular categories. We will also present some constructions of modular categories such as gauging and zesting.

Within the category of modules over a ring R, finiteness may refer to the study of conditions that R must fulfill in order to have a finite degree of coherency. In particular, this involves the question of whether a given ring is noetherian or coherent. In this talk, my goal is to revisit some important developments in the theory of finiteness conditions, pointing out recent findings I have with collaborators from Latin American institutions. Recall that a ring R is (left) coherent if every finitely generated (left) ideal of R is finitely presented. In terms of modules over R, the latter is equivalent to saying that for every finitely presented R-module M there exists an exact sequence of the form F2 → F1 → F0 → M → 0 where Fk is a finitely generated and free R-module for k = 0, 1, 2 (that is, M has a finite presentation of length 2). A more general notion can be traced back to Bourbaki, namely, modules of type FPn. These are R-modules with a finite presentation of length n. This concept is useful to propose a more general notion of coherency for rings. Namely, a ring R is n-coherent if, and only if, every R-module of type FPn is of type FPn+1.

Some recent developments have established characterizations of n-coherent rings in terms of homological closure properties of the class FPn(R) of R-modules of type FPn, as well as of its corresponding classes of injective and flat R-modules relative to FPn(R), denoted by FPn-Inj(R) and FPn-Flat(R). To mention one, a ring R is n-coherent if, and only if, FPn-Inj(R) is the right half of a hereditary cotorsion pair. Moreover, the classes FPn(R), FPn-Inj(R) and FPn-Flat(R) have other important features that allow us to propose and characterize another important family of rings, the n-hereditary rings. As a matter of fact, when FP1-Inj(R) is a torsion class (or equivalently, when R is semi-hereditary) one has a good control of pure exact sequences of R-modules. I will present some of these advances in the context of modules, along with their corresponding analogs in more general settings such as Grothendieck categories. I will also mention an important open question, which claims the possibility of characterizing n-coherent rings only in terms of finiteness conditions on its ideals (in a similar way as in the concept of coherent ring).

The deformation theory of algebras was initiated by Gerstenhaber in the 1960s. In this talk we will show the connection between deformations of algebras and their Hochschild cohomology, and we will also consider the description of deformations via the Maurer-Cartan equation and differential graded Lie algebras. We will explain how the category of modules of a deformed algebra can be described in terms of the cocycle selected and the category of modules of the original algebra. Several examples will be discussed.

Let f be a real polynomial with n variables and let X be a real algebraic set.

We ask (1) How to find the global minimum of f on X? and perhaps more importantly: (2) How to do this effectively? focusing on quantifying the magnitude of the "computational resources" needed to do so.

In this talk I will briefly review a "sum-of-squares" approach to these problems and discuss several novel connections between real and complex algebraic geometry that arise from trying to measure the complexity of sum-of-square nonnegativity certificates on varieties of low dimension.

Let A be a finitely generated domain over a field of characteristic p > 0. If A is graded by an abelian group M, then the integral closure of A is also M-graded when M is p-torsion free. But this fails when M has p-torsion, for example when M = Z/pZ and A = k[x,y]/(y^p - f(x)), where x has degree 0, y has degree 1 and f is a "general" polynomial. More generally, an action of a finite group scheme on an algebraic variety need not lift to an action on its normalization. The talk will discuss a notion of equivariant normalization that remedies this problem. As applications, we will present a Hurwitz formula for inseparable covers of smooth projective curves, and a classification of smooth projective surfaces whose automorphism group contains an elliptic curve.

In this talk, I will discuss two different symmetric function generalizations of the chromatic polynomial. The famous classical one is Richard Stanley’s chromatic symmetric function, which has led to a great deal of activity in combinatorics and algebraic geometry. In particular, I will recall a refinement, introduced in my work with John Shareshian, of the Stanley–Stembridge e-positivity conjecture and its connection to Hessenberg varieties. The other generalization was introduced more recently in my work with Rafael González D’Léon on weighted bond lattices. An e-positivity result and conjecture for these non-Stanley chromatic symmetric functions will also be discussed.

We will discuss representations of the symmetric group and the general linear group, including some long-standing fundamental conjectures about them and how they were eventually proved incorrect. We will then explain a new conjecture (now a theorem) and explore its surprising connections to fractals and billiards.

Let Lambda be an Artin algebra and K^b(Proj(Lambda)) be the triangulated category of bounded cochain complexes in Proj(Lambda). It is well known (Adachi-Iyama-Reiten, 2014) that two-term silting complexes in Proj(Lambda) are described by using the tau-tilting theory in Module(Lambda).

In this talk, we give a characterization of certain n-term silting complexes in Proj(Lambda) which are induced by Lambda-modules. In orden to do that, we introduce the notions of tau_n-rigid, tau_n-tilting and tau_{n,m}-tilting Lambda-modules. The later one is both a generalization of tau-tilting and tilting in Module(Lambda). We also state some variant, for tau_n-tilting modules, of the well known Bazzoni's characterization of tilting modules.

We also give some connections between (n+1)-term silting complexes in the triangulated category K^b(Proj(\Lambda)) and tau_n-rigid modules in Module(Lambda). Moreover, a characterization are given to know when a tau_n-tilting module is n-tilting. It is also presented the properties of the tau_{n,m}-tilting modules, in particular, of being m-tilting in some quotient algebras. We apply the developed tau_{n,m}-tilting theory to the finitistic dimension conjecture, and thus, for the particular case m=n=1, we get a result obtained by P. Suarez (2021). If the time allows us, we close this talk by stating and discussing some open questions (conjectures) that we consider crucial for the future develop of the tau_{n,m}-tilting theory.

In recent years there has been a great deal of interest in detecting properties of the fundamental group of a 3-manifold via its finite quotients, or more conceptually by its profinite completion.

This motivates the study of the profinite completion of the fundamental group of a 3-manifold.

I shall discuss a description of the finitely generated prosoluble subgroups of the profinite completions of all 3-manifold groups and of related groups of geometric nature.

Given a discrete group G and a family F of subgroups of G there exists a G-CW complex that classifies G-CW complexes with isotropy contained in the family F. Such space is unique up to G-equivariatn homotopy and it is called the _classifying space of G for the family F_. For the trivial family, it is the universal covering space of a K(G,1)-space and, for the family of finite subgroups, it is the universal space for proper actions of G. More generally, classifying spaces for families play an important role in the classification of manifolds with a given fundamental group, and having geometric models of small dimension can be useful for computations. In this talk, we will introduce these notions and survey what is known about classifying spaces for some natural families of subgroups of the mapping class group of a surface of finite type. When finding explicit geometric models is out of reach, we show how algebraic methods can be used to compute, or at least bound, the minimal dimension of such classifying spaces.

In this talk, we will present some examples of the interaction between Algebra and Geometry. Focus on three different objects: Riemann surfaces, Abelian varieties, and linear representations of finite groups. They all give rise to deep theories with their own known open problems, and considerable steps toward their understanding arise from their merging. We will explore some of these lines of interaction and present an effective procedure to explicitly find the decomposition of a polarized abelian variety into its simple factors if a period matrix is known. In addition, we will show two algorithms to compute the period matrix for an abelian variety, depending on the given geometric information about it. The goal is to fully decompose an abelian variety with a non-trivial automorphism group by successively decomposing their factor subvarieties arising from the group action, even when these no longer have a group action. We will also illustrate how to use our algorithms, showing a completely decomposable Jacobian variety of dimension 101, which fills the Ekedahl-Serre gap.