This talk will focus on two problems at the interface of geometry and topology.

The first problem is inference of conditional dependencies given observations from a multivariate probability density. This is the problem of learning structure in a graphical model. We develop a parameterization of hypergraphs based on the geometry of points in d-dimensions. Informative prior distributions on hypergraphs are induced through this parameterization by priors on point configurations via spatial processes. The approach combines tools from computational geometry and topology with spatial processes and offers greater control on the distribution of graph features than Erdos-Renyi random graphs.

The second problem develops a topological approach to stratification for point cloud data. Given point cloud data drawn from a stratified space we provide an algorithm to infer which points belong to the same strata. First we define a multi-scale notion of a stratified space, giving stratifications over a set of resolution levels. We then use methods derived from kernel and cokernel persistent homology to cluster the data points into different strata, and we prove a result which guarantees the correctness of our clustering, given certain topological conditions.

Sayan Mukherjee is an Assistant Professor in the Departments of Statistical Science, Computer Science, Mathematics and the Institute for Genome Sciences & Policy. His research foci include geometry and topology in statistical inference, Bayesian models for high-dimensional data analysis, and applications in computational biology. He did his PhD from the AI Lab at MIT and was a PostDoctoral Fellow at The Broad Institute of MIT and Harvard.