Abstract: In this joint work with Joaquin Rodrigues Jacinto, we develop the theory of locally analytic representations from the perspective of Condensed mathematics of Clausen-Scholze. Taking as inspiration foundational works on the subject from Lazard/Schneider - Tetelbaum/Emerton, etc., we reprove and generalize some cohomological comparisons between continuous, locally analytic and Lie algebra cohomology.

Abstract: Algorithms in data science often take parameters, which are chosen using rules of thumb or by making strong assumptions about the data. Persistence theory studies parametrized families of topological spaces by means of algebraic objects called persistence modules and, by describing the structure of persistence modules, it provides us with topological descriptors that summarize the varying topology of a parametrized family of spaces. In particular, persistence can be used to summarize the output of many geometric algorithms as we vary their parameters, and thus provides novel tools for parameter selection. It was recognized early in the development of persistence that the structure of persistence modules with two or more parameters -- called multiparameter persistence modules -- is significantly more complicated than the structure of their one-parameter counterparts and much effort has gone into the development of descriptors for multiparameter persistence modules. I will describe two related research programs of mine: applications of multiparameter persistence to density-based clustering, and the development of visualizable and stable descriptors of multiparameter persistence modules using tools from relative homological algebra.

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