Marta Casanellas

Universitat Politècnica de Catalunya

Algebraic geometry for phylogenetic trees and networks

Phylogenetics is the area of biology that studies species' evolutionary relationships from their molecular sequences. These relationships are represented on a phylogenetic tree or network. Modeling nucleotide or amino acid substitution along a phylogenetic tree via a Markov model is one of the most common approaches in phylogenetic reconstruction. Then a Markov process on a phylogenetic tree or network parametrizes a dense subset of an algebraic variety, the so-called phylogenetic variety.
During the last two decades, algebraic geometry has been used for reconstructing phylogenetic trees and for establishing the identifiability of parameters of complex evolutionary models (and thus guaranteeing model consistency). In this talk, we will explain how to find equations in the ideal of phylogenetic varieties and how these equations can be used for phylogenetic reconstruction. We will also explain the role of these equations in the identifiability of phylogenetic networks.

Pedro Macias Marques

Universidade de Évora

Cohomological blowups of graded Artinian Gorenstein algebras

Given graded Artinian Gorenstein (AG) algebras \(A\) and \(T\), and a surjective map \({\pi: A \to T}\), we can construct a new AG algebra, mimicking the behaviour of cohomology rings of a blowup of a complex manifold along a submanifold.
In joint work with Tony Iarrobino, Chris McDaniel, Alexandra Seceleanu, and Junzo Watanabe, we study this construction from an algebraic point of view, looking at its Hilbert function, Macaulay dual, and how it preserves the strong Lefschetz property.

Alejandro Calleja Arroyo

ICMAT - UCM

Character Varieties of Knots

Given an algebraic group \(G\) and a knot \(K\subset \mathbb{S}^3\), we define the \(G\)-character variety of \(K\) as the moduli of representations \(\rho:\pi_1(\mathbb{S}^3-K)\to G\) of the knot group into \(G\). The importance of these varieties lies in the fact that their study provides in a natural way many knot invariants. In this talk, we will introduce one of the most important of these invariants, the \(E\)-polynomial, exposing the techniques used to study them, as well as the main results known, focusing on the case of torus knots.

Erroxe Etxabarri Alberdi

Mondragon Unibertsitatea

1-dimensional components of the K-moduli of Fano 3-folds

We introduce \(K\)-stability, and the motivation behind it. We will see how to study and completely describe all one-dimensional components of the \(K\)-moduli of smooth Fano 3-folds. And we will finish giving some specific examples for family 3.12 (blow-up of a disjoint line and twisted cubic on \(\mathbb{P}^3\)). This result is in collaboration with Abban, Cheltsov, Denisova, Kaloghiros, Jiao, Martinez-Garcia and Papazachariou.

Enric Florit

Universitat de Barcelona

Abelian varieties genuinely of \(\mathrm{GL}_n\)-type

The representations of the absolute Galois group arising from abelian varieties are a central object of study in arithmetic geometry. Classically, one- and two-dimensional such representations have been considered. In this talk, I will introduce the definition of an abelian variety of \(\mathrm{GL}_n\)-type. This notion essentially tells us the dimension of the Galois representation attached to such a variety, and it is usually too coarse for applications. In order to refine it, we will see a characterization of those abelian varieties with a symplectic or an orthogonal Galois representation, which is given in terms of the geometric endomorphism algebra. This is joint work with Francesc Fité and Xavier Guitart.

Francisco García Cortés

Universidad de Sevilla

Not fundamental groups

Esnault and de Jong found an arithmetic obstruction for a group to be the topological fundamental group of a smooth quasi-projective complex algebraic variety. After reviewing it, a family of groups for which this obstruction takes place will be presented. If time permits, we will see that the employed methods allow us to understand the dimension of the \(SL2\)-character varieties associated to general 2-generated 1-relator (of length 4) groups. Joint work with Benjamin Church (arXiv:2410.23233).

Irene Macías Tarrío

Universitat de Barcelona

Brill-Noether Theory of stable bundles on ruled surfaces

Let \(X\) be a smooth projective surface over an algebraically closed field \(K\) of characteristic 0 and \(H\) an ample divisor on \(X\). Consider \(M_H =M_{X,H} (r; c_1,c_2 )\) the moduli space of \(H\)-stable rank-r vector bundles on \(X\) with fixed Chern classes \(c_i(E):=c_i\) for \(i=1,2\). One way to study the geometry of the moduli space \(M_H\) is by considering its subvarieties. In particular, one can consider the subvarieties called Brill-Noether loci, whose points are stable vector bundles having at least \(k\) independent sections. In this talk, we will focus the attention on study the non-emptiness and the geometry of Brill-Noether locus in the case when \(X\) is a ruled surface.

Josep Pérez Díez

Universitat de Barcelona

Betti numbers of full Perazzo algebras

It is well-known that any Artinian Gorenstein algebra \(A\) is associated to a homogeneous polynomial via Macaulay duality. This polynomial is called Macaulay dual generator for \(A\). In this talk, we provide a geometric characterization of full Perazzo algebras, whose Macaulay dual generator is a form of degree \(d\), in \(n+1+m\) variables, such that \(n+1=\tbinom{d+m-2}{m-1}\). We will show that a full Perazzo algebra is the doubling of a zero-dimensional scheme \(Z\subset \mathbb P^{n+m}\), consisting of \(n+1\) double points. From this characterization we will obtain the graded Betti numbers of such algebras.

Miguel Recio Ibáñez

Universidad Complutense de Madrid

Criterios cohomologicos en grassmannianas a partir de regularidad

El Teorema de Horrocks permite caracterizar los fibrados vectoriales en la recta proyectiva que son suma directa de fibrados lineales como aquellos que cumplen ciertas anulaciones cohomologicas; en particular, dice que ser suma directa de fibrados lineales es equivalente a no tener cohomología intermedia. Existen muchas demostraciones del Teorema de Horrocks; una que nos interesa particularmente utiliza las propiedades de la regularidad, en el sentido de Castelnuovo-Mumford, de haces coherentes sobre la recta proyectiva. Costa y Miró-Roig introdujeron un concepto de regularidad para variedades lisas que cumplen ciertas condiciones. Esta regularidad tiene las propiedades necesarias demostrar un análogo al Teorema de Horrocks para este contexto más general. Exploraremos, pues, cómo aplicar estas herramientas para obtener caracterizaciones por medio de anulaciones cohomologicas de fibrados vectoriales en grassmannianas como sumas directas de fibrados lineales y universales.

Adina Veronica Remor

Universidade Federal do Paraná

Regularity of extensions of Noetherian local rings

There is a strong connection between regularity and homological algebra, particularly established by the Serre-Auslander-Buchsbaum theorem. In 1956, as a generalization of homological algebra, Hochschild introduced the notion of relative homological algebra. It is natural to ask whether a relative version of the theorem holds, using relative global dimension. This question is the focus of the present work, which is a joint project in progress with K. Iusenko, V. Pretti and R. M. Miró-Roig.
RGAS Universitat de Barcelona IMUB