Shouda Wang: Equality cases of geometric and combinatorial inequalities: from Alexandrov–Fenchel to generalized Mason

Abstract: Inequalities in convex and algebraic geometry have led to numerous recent developments in log-concavity results in matroid theory. At the core of these developments is the Alexandrov–Fenchel (AF) inequality for Lorentzian polynomials. While classical geometric inequalities such as the isoperimetric and Brunn–Minkowski inequalities have simple equality conditions, the AF inequality, even when restricted to polytopes, has a gigantic number of nontrivial equality cases, which are provably hard to describe in the sense of complexity theory.

In this talk, I will first recall the equality characterization of the AF inequality for polytopes and then explain how much of this machinery extends to general Lorentzian polynomials. I will then explain how these tools can be applied to the equality cases of the generalized Mason inequality. This is based on joint work with Shiqi Cao.