Jonathan Leake: Transportation polytope volume bounds via polynomial capacity and Lorentzian polynomials

Abstract:
Polynomial capacity has been used in the past 20 years to obtain lower bounds on various combinatorial quantities, including the permanent, the mixed discriminant, matchings of a graph, and the intersection of two matroids (to name a few). More recently, Lorentzian polynomials have been used to derive log-concavity statements, e.g. on the independent sets of a matroid or on the coefficients of Schur polynomials. In joint work with Petter Brändén and Igor Pak, we have recently combined these two to obtain improved lower bounds on the number of contingency tables with given marginals. Contingency tables can be viewed as the lattice points of transportation polytopes and flow polytopes more generally, and so these lower bounds imply volume lower bounds for such polytopes. In this talk, we discuss these bounds along with a few key points of their proofs.

Zoom meeting ID: 654 5562 3260

Zoom link: https://kth-se.zoom.us/j/65455623260