Shayan Oveis Gharan

University of Washington

February 25, 2019

A matroid is an abstract combinatorial object which generalizes the notions of spanning trees,

and linearly independent sets of vectors. I will talk about an efficient algorithm based on the Markov Chain Monte Carlo technique

to approximately count the number of bases of any given matroid.

The proof is based on a new connections between high dimensional simplicial complexes, and a new class

of multivariate polynomials called completely log-concave polynomials. In particular, we exploit a fundamental fact from our

previous work that the bases generating polynomial of any given matroid is a log-concave function over the positive orthant.

Based on joint works with Nima Anari, Kuikui Liu, and Cynthia Vinzant.