Title: Differentiable Topological Data Analysis

Abstract: Topological Data Analysis is concerned with the study of the sublevel-sets of functions defined on data. Such objects can be modeled as persistence modules, i.e., functors from  products of totally ordered sets to the category of vector spaces. One of the key properties of persistence modules is their stability under perturbations of their originating functions: this not only validates their use in data analysis contexts, where they are robust to noise in the data, but it also makes it possible to do differential calculus and optimization with them, thus enabling new  applications for instance in deep learning. In this talk I will give a brief introduction to persistence modules, then I will explain how one can equip their category with what looks like a differential structure, to do differential calculus and optimization. My focus will be primarily on the special case where the poset is totally ordered, which appears in many applications and where persistence modules are simpler to work with because they decompose into elementary summands. If time permits I will also touch upon the general setting, in which the structure of persistence modules is significantly more complex and difficult to work with.