Alejandro Morales, Université du Québec à Montréal

Title: Bounds on integral flows of graphs and the Kostant partition function

Abstract: Integer flows on networks are very important objects in optimization, combinatorics, and representation theory. For example the number of integer flows on a complete graph is also known as the type A Kostant's vector partition function that can be used to compute Kostka and Littlewood-Richardson coefficients. In this talk we will talk about bounds on the Kostant partition function starting from elementary ones, continuing with more sophisticated ones using subdivisions of flow polytopes, and ending with lower bounds based on Gurvits capacity method of certain Lorentzian polynomials. The last method adapts the results of Brändén--Leake--Pak on lattice points of transportation polytopes to flow polytopes whose lattice points are counted by the Kostant partition function.

The talk is based on joint work with Jonathan Leake.