Sayan Mukherjee : Stochastic Topology

One of the exciting results of applied algebraic topology for data analysis has been a formulation of the field of "Stochastic topology." This is an intersection of topology and probability/statistics. I will present some research directions in this field: 1) Euler integration for stochastic models of surfaces and shapes: how topological summaries such as persistence homology or Euler characteristics curves can be used to model surfaces and measure distances between bones. 2) Random simplicial complex models: Given m points drawn from a distribution on manifold construct the union of balls of radius r around these points. As m goes to infinity and r goes to zero what can we say about the limiting distribution of Betti numbers or critical points of this random process ? 3) Spectral theory of simplicial complexes: There is a well developed spectral theory for graphs that provides insights on random walks, spectral clustering of graphs, and near linear time algorithms for solving a system of linear equations. How do these ideas extend to simplicial complexes, in particular: a) is there a notion of a Cheeger inequality for clustering to preserve holes ? b) how does one define a random walk on simplicial complexes that have limiting distributions related to the Harmonics of the (higher order) Hodge Laplacian ? c) we conjecture that the question of near linear time algorithms for linear systems is related to a notion of discrete Ricci curvature for graphs. I just expect knowledge of basic math and will focus on motivating concepts rather than details.

• Category: Graduate/Faculty Seminar
• Duration: 01:34:43
• Date:  December 5, 2014 at 4:25 PM
• Views: 115
• Tags:   seminar, Graduate/faculty Seminar