Matías Raja

My research focuses on the geometry and topology of Banach spaces, but sometimes the problems considered there have led me to other areas such as general topology, descriptive set theory, locally convex spaces, optimization... I will describe my mathematical production in more detail, grouped into several lines, although interaction or overlap between them is inevitable: - Renorming theory, which consists of relating the existence of equivalent norms with special characteristics (convexity, smoothness, topological) with other properties of Banach spaces, often of a very different nature. In this field, in my Ph.D. dissertation, I invented a technique for constructing norms from countable decompositions that allowed me to isomorphically characterize the existence of equivalent Kadets norms, as well as to simplify and extend the results on locally uniformly convex renorming. The technique, with various modifications, has subsequently been used by other authors such as A. Moltó, J. Orihuela, M. Valdivia, S. Troyanski, S. Lajara, and R. Smith. In this line, I would highlight our work on uniformly convex renorming, which has been cited by G. Pisier in his book "Martigales in Banach spaces." - Convexity in infinite dimensions: properties of convex sets, extreme points, exposed points, Choquet theory, Radon-Nikodym property, convex functions, ordinal indices... As an example, one of my most significant results combines the Krein-Milman theorem with the Baire property: a lower bounded and semicontinuous convex function defined on a compact convex set of a locally convex space is continuous at some extreme point. Regarding ordinal indices, we have used variants of the Szlenk index to characterize, for example, functions that can be uniformly approximated by differences of convex Lipschitz functions. - Descriptive set theory, particularly Borel sets in non-metrizable and non-separable spaces (among which the prototype is a Banach space with the weak topology) and the measurability of functions in this same context. In my PhD dissertation, I introduced a method for the classification of Borel sets in a non-metrizable context, which has been accepted by other authors, and I showed that Borel subsets of a compact set are Borel in any immersion in a superspace. The relationships between measurability and linear structure have been exploited in some works, extending a result by J. Saint-Raymond, for instance. - General topology, particularly that related to the properties of weak or weak* topologies: metrizibility, covering properties, compact classes related to Banach spaces (Eberlein, fragmentable, projective, etc.), spaces of continuous functions, and Radon measures. In this regard, the contribution I would highlight is the introduction of Namioka-Phelps compacts, used by R. Haydon as an ingredient in a deep renorming result, and the study of projective compacts. - Weak supercompactness, which is a notion of compactness for subsets of a Banach space intermediate between norm and weak compactness, which, in a sense, localizes the idea of superreflexivity. The theory of this type of sets and their applications has been largely developed in our work, some of it in collaboration with G. Lancien of the Université Franche-Comté (Besançon) and Guillaume Grelier, my most recent PhD student. More information about my research and academic work can be found on the corporate website (link above). Scientific dissemination and interesting facts can be found on my personal website https://matiasraja.es