Geometric central limit theorems on stratified spaces

The central limit theorem (CLT) is commonly thought of as occurring on the real line, or in multivariate form on a real vector space. Motivated by statistical applications involving nonlinear data, such as angles or phylogenetic trees, the past twenty years have seen CLTs proved for Fréchet means on manifolds and on certain examples of singular spaces built from flat pieces glued together in combinatorial ways. These CLTs reduce to the linear case by tangent space approximation or by gluing. What should a CLT look like on general non-smooth spaces, where tangent spaces are not linear and no combinatorial gluing or flat pieces are available? Answering this question involves figuring out appropriate classes of spaces and measures, correct analogues of Gaussian random variables, and how the geometry of the space (think "curvature") is reflected in the limiting distribution. This talk provides an overview of these answers, starting with a review of the usual linear CLT and its generalization to smooth manifolds, viewed through a lens that casts the singular CLT as a natural enhancement, and concluding with how this investigation opens gateways to further advances in geometric probability, topology, and statistics. Joint work with Jonathan Mattingly and Do Tran.