16:00 – 17:00
Densities for Hausdorff measure and rectifiability. The Besicovitch 1/2 conjecture
Jaume Capdevila (UB)
In this work we study two central concepts of geometric measure theory: densities for Hausdorff measure and rectifiable sets. In particular, we focus on a specific aspect of the relationship between these two concepts, the Besicovitch 1/2 conjecture. We present the theory developed by Besicovitch in his pioneering papers published in 1938 and 1939, to prove the characterization of rectifiability in the plane in terms of densities. We also study the paper by Preiss and Tišer published in 1988, which improved previously known results on the conjecture. Finally, we present two original contributions. First, we generalize to Euclidean space ℝn an example of a set of points in the plane originally constructed by Besicovitch and prove its main properties, thus extending a lower bound of the conjecture to arbitrary dimension. Second, we prove using only densities that if a set E in the plane is purely 1-unrectifiable, then the Cartesian product E×J with an interval J is purely 2-unrectifiable, provided the Besicovitch 1/2 conjecture is assumed true.
Exploring the principles of coexistence in the invader-driven replicator equation“
Marina Garcia (UPC)
The replicator equation, originally introduced in evolutionary game theory, has been widely applied in biology to model the complex dynamics of systems composed of many species, such as multispecies ecological communities or multistrain microbial pathogens. In this work, we use the replicator equation to explore one of the fundamental questions of evolutionary biology and ecology, which is how biodiversity is generated and maintained, focusing on invader-driven systems, where the interactions or fitnesses of species are defined by the invading species, independently of the invaded species. With the aim of relating fitnesses to the identities and number of surviving species at equilibrium states, we find the mechanism governing the selection of the final set of surviving species through numerical simulations and analysis, which leads to the maximization of the system's resistance to external invasions.
Unique preduals and free objects in Banach spaces
Mario Guillén (UV-UPV)
We investigate when a Banach space has a unique Banach predual. First, we explore the existence and uniqueness of preduals in various types of spaces. In the case of Lipschitz function spaces, we review current results and outline the proof of the uniqueness of the predual. Next, we revisit two known conditions that guarantee uniqueness: being (separably) «L-embedded» and possessing property (X). The central part of the work focuses on the classical space of bounded holomorphic functions on the unit disk. After analyzing Ando's proof establishing the uniqueness of the predual of this space, we extend Ando's result to the case of a union of open, disjoint, and simply connected subsets, and explore various strategies to extend it to the case of bounded holomorphic functions of several complex variables, explaining why our approach to proving it is difficult.
Density of hyperbolicity in families of complex rational functions
Francesc Timoner (UB)
In this work, we address the fundamental open problem of whether hyperbolic rational functions, those for which every critical point lies in the basin of an attracting cycle, are dense in the space of rational functions of the same degree. By this, we mean whether any such function can be uniformly approximated on compact sets by hyperbolic functions. Conjecturally, the answer is «yes», and this is known as the Density of Hyperbolicity Conjecture. After reviewing key tools from complex dynamics such as puzzle constructions, quasiconformal conjugacies, Böttcher coordinates, and holomorphic motions, we introduce complex functions as a natural extension of polynomial functions and discuss their rigidity under combinatorial equivalence. Focusing on non-renormalizable polynomials without neutral periodic points, we reproduce, clarify, and verify the result of Kozlovski–van Strien that these polynomials admit approximation by hyperbolic functions by constructing dynamically natural box functions and applying topological and rigidity results. In conclusion, we will outline how this framework, with careful adjustment, promises to extend beyond the polynomial case to prove density of hyperbolicity in broader families such as Newton and McMullen functions, thus outlining a clear path for future advances in complex dynamics.
17:30 – 18:30
Fusion theorems and applications
Luis Pablo Colmenar (UV-UPV)
In finite group theory, many important results are expressed in terms of Sylow subgroups and are proved using the classical Sylow theorems. Faced with this type of result, it is natural to ask whether they can be extended or generalized. One natural direction is to replace Sylow subgroups with Hall subgroups. In this talk, we will explore how a well-known result of Wielandt can become a powerful tool in this context. We will present two main applications: one related to Alperin's fusion theorem, an essential result in the study of fusion in finite groups, key to addressing local-global type problems, and another focused on a less known concept, the subnormalizer, studied mainly by Carlo Casolo. The latter has connections with current conjectures in character theory and offers a new perspective on the interaction between group structure and representation theory.
Topology of Complex Polynomials
Manuel García (UV-UPV)
Since 1983, with the work of Broughton, several regularity conditions at infinity for the polynomial f have been introduced that guarantee the absence of critical values at infinity. In this work, we collect the most relevant regularity conditions and study the relationships between them. In particular, we answer two open questions proposed by Lê Dũng Tráng and J.J. Nuño-Ballesteros in: Lê, D.T., Nuño Ballesteros, J.J., A remark on the topology of complex polynomial functions, RACSAM 113, 3977–3994 (2019). Finally, we also review a global version of the Milnor Sphere Bundle Theorem.
Idempotent elements of the group algebra
Vicent Miralles (UV-UPV)
This work focuses on the study of idempotent elements of the group algebra, with a particular emphasis on centrally primitive idempotents. These elements are fundamental because they allow the algebra to be decomposed into simpler blocks. The importance of centrally primitive idempotents lies in the fact that each one generates one of these blocks, and moreover, they form a basis of the center of the algebra, which completely defines its structure.
The main objective is to develop an explicit and practical method for calculating these idempotents over fields whose characteristic does not divide the order of the group (which we assume to be finite), and which are often not algebraically closed. This is not an easy task, as many results from representation theory rely on this latter property and are not valid in a more general context. For this reason, we resort to the concept of a splitting field for a group, which generalizes the algebraically closed field by providing a theoretical framework that guarantees the validity of many classical results, including the expression of these idempotents.
The method we develop, often known as Galois descent, consists of exploiting the expression of the centrally primitive idempotents of the group algebra over a splitting field. The idea is to consider an extension of the original field that is a splitting field for the group, and in this extension, the Galois group of the extension acts on these idempotents. The expression of these idempotents is known since they are defined over a splitting field; thus, it is shown that the sum of the orbits resulting from this action finally provides us with the centrally primitive idempotents we seek in the original field. This method for obtaining these idempotents is not the only one, but it is notably simpler than other methods.
The work is structured into five chapters. The first consists of a review of the theoretical foundations of algebras, modules, group representations, and characters. The second introduces the necessary tools, such as the tensor product, to formalize field extensions. Furthermore, splitting fields for a group are studied, proving the uniqueness of the decomposition into homogeneous components of the group algebra associated with these fields and the expression of the projections onto these components, closely related to centrally primitive idempotents. Next, the third chapter defines idempotents, explores their relationship with the decomposition of the group algebra into ideals, as well as with projections. In the fourth chapter, the Galois descent method is developed in depth, which uses the action of the Galois group on representations, modules, and their idempotents to obtain the objective result of the work. Finally, the fifth chapter shows an example of the application of all the developed theory for the concrete case of a finite field.
On nilpotency in braces and the Yang-Baxter equation
Alberto Rodríguez (UV-UPV)
Braces are an algebraic structure that allows us to study the set of non-degenerate solutions of the Yang-Baxter equation (YBE). Every brace admits a non-degenerate solution of the YBE, and conversely, every non-degenerate solution of the YBE is controlled by the solution associated with a brace of the solution. In this way, the problem of classifying non-degenerate solutions undoubtedly passes through the structural study of braces. Thus, algebraic properties of braces translate into properties of solutions and vice versa. In particular, notions of nilpotency of braces allow characterizing the multipermutational character of braces, one of the classes of solutions that receives the most attention in the theory.
In this work, we address the concepts of nilpotency of braces and their relationship with solutions. We analyze their local versions of p-nilpotency. In particular, we present an original contribution to the theory, introducing the analysis of the pcentral nilpotency of a brace and its associated p-Fitting ideal.