Description:

Title: Volume Inequalities for Convex Bodies, Matrices, and Divisors

Abstract: Classical inequalities such as the Loomis-Whitney and Alexandrov-Fenchel inequalities reveal deep geometric properties of convex bodies. In this talk, we explore their analogs in two parallel settings: Hermitian matrices and divisors on projective varieties. We introduce the notion of volume polynomials associated to convex bodies, Hermitian matrices, and divisor classes, and investigate the relationships among them. These connections lead to new inequalities for mixed volumes, mixed discriminants, and intersection numbers. This is joint work with June Huh, Mateusz Michałek and Jian Xiao.