"Homotopy groups in formulas of S1S" by Antimateria

Abstract

There have been many attempts at finding a topological invariant for automatas. In this talk I will apply two particularly simple and naive invariants to formulas in Monadic Second-order Logic of One Successor (S1S) using the Büchi automaton associated with them. The invariants presented are not particularly powerful, but they are a good example of what a topological invariant can look like in a logical theory.

"Model theory: type spaces and the Ryll-Nardzewski theorem" by Sergio Esperalta Delgado

Abstract

In this talk I will give the definition of types and the topological spaces arising from them, one of the building elements widely used in Model Theory, and how they can be used to prove some results about א-null categorical theories. The aim will be foundational, and some equivalences and reinterpretations will be presented for the sake of understanding. 

"An invitation to differential algebra" by Jorge Carrasco  Coquillat

Abstract 

We introduce differential rings and state results analogous to those of commutative algebra, such as the Ritt-Raudenbush basis theorem (analogous to Hilbert's basis theorem) or the decomposition theorem. We take a look at differentially closed fields and hint at their importance by introducing the differential Nullstellensatz. Finally, we introduce Schanuel's conjecture, which is today a central topic in applied model theory.

"Interactions between Topology and Combinatorics" by Juan Martín Fajardo 

Abstract 

Many surprising connections exist between topology and the theory of computation. This talk explores one such connection: the application of topology to Constraint Satisfaction Problems (CSPs). We will survey recent developments in using topological formalism to analyze the complexity of certain CSPs. In particular, we focus on graph coloring and examine the proof of the Hell-Nešetřil theorem by J. Opršal and S. Meyer.

"Equivariant algebraic geometry" by Javier Herrero Cañedo

Abstract 

In this talk, I will give an introduction to the basic concepts of equivariant algebraic geometry in the affine case (i.e. the theory of Hopf algebras and their comodules). I will also discuss the concept of quotient stacks and connections to geometric invariant theory. I will conclude relating all of this to equivariant derived categories and window subcategories, and how this recovers the classical SOD of projective space.

"An introduction to Higher Algebra" by Pedro Mayorga Pedraza

Abstract 

In this talk we introduce the basic ideas of homotopy theory and higher category theory with the intention of doing algebra in this new world. We thus ponder on very basic algebraic concepts as associativity, commutativity and group completions and discuss what is necessary to extend them to the context of higher algebra.

"The Landsberg-Berwald conjecture in semi-Finsler geometry" by Francisco Manuel Muñoz Muñoz

Abstract 

The lecture will start from the generalization of Minkowski norms. Special emphasis is placed on the transition from a classical framework (positive-definite) to the general non-degenerate case. Due to the intrinsic directional dependence of the metric, the geometric development is grounded in the pullback bundle. Over this space, the four classical affine connections (Cartan, Berwald, Chern and Hasiguchi) are systematically derived and related with a new emergen framework: Anisotropic connections. Finally, this formalism is applied to address the Landsberg-Berwald conjecture, analyzing the structural conditions under which a Landsberg space trivially reduces to a Berwald space. 

"The magic theorem" by Miguel de Navarro y Muñoz

Abstract 

In this talk I classified the 17 groups of symmetry you can find on a wallpaper using the orbifolds they define. For that purpose we studied orbifolds and their Euler characteristic with a bunch of poorly hand-drawn examples. Finally, I briefly grazed geometric structures on orbifolds and proved by example how to construct tilings of the sphere, the Euclidean plane and the hyperbolic plane from the corresponding quotient orbifold.

"Revisiting Serre Duality in Derived Categories" by Alejandro López Salazar

Abstract 

In complex geometry, Serre duality can be described in an analytic way. Would it be possible to study this notion in a more general context, using other tools? In this talk, Serre duality is presented in an algebraic language, introducing triangular categories, derived categories and basic notions of this tool on coherent sheaves. Finally, a revisited formulation of Serre duality is presented.

"Six-functor formalisms" by Jordi Cardiel

Abstract 

Given a nice category of geometric objects, we usually have operations (such as (exceptional) pullbacks and (exceptional) pushforwards, and tensor products and internal homs) that satisfy certain compatibilities (these define pairs of adjunctions and, for example, satisfy proper base change, Künneth formulas, Poincaré/Verdier dualities and projection formulas). The aim of this talk is to motivate and introduce the notion of a 6-functor formalism (following Heyer-Mann) which provide a general framework in which these ideas live.

"Regularization for linear regression" by Jaime Díaz-Trechuelo Sánchez-Moliní

Abstract 

In this talk, we motivate the need for regularization techniques in regression. We first examine the challenges posed by highly correlated predictors, limited interpretability, and ill-conditioned estimation problems, and introduce regularization as a framework for addressing them. We then present ridge regression and discuss how strict convexity helps restore well-posedness and numerical stability. Next, we introduce the Lasso and use subdifferential calculus to rigorously explain its variable-selection property. We conclude with some remarks on the elastic net, which combines the main strengths of ridge regression and the Lasso.

"Fixed points and KAM theory" by Daniel Sánchez-Simón del Pino

Abstract 

In this talk, we will present the most basic aspects of KAM theory using an academic example. In this manner, we show how the theory developed by Kolmogorov, Arnold, and Moser allows us to prove fixed point results in linearized problems where small denominator problems arise. 

"Introduction to Gromov-Hausdorff convergence" by Diego Jiménez Téllez

Abstract 

This talk introduces the Gromov-Hausdorff distance, a fundamental metric defined on the space of isometry classes of compact metric spaces. By removing the requirement of a fixed ambient space, this convergence provides an intrinsic framework for taking geometric limits. It serves as the bedrock of modern metric geometry, underpinning Gromov's Precompactness Theorem for Riemannian manifolds under curvature and diameter bounds. Ultimately, this framework extends geometric analysis beyond smooth spaces, providing the rigorous machinery required to study metric collapse and the emergence of singular limit spaces where classical topological invariants fail.