Multiplicities and Volumes: An interplay among Algebra, Combinatorics, and Geometry

The notion of multiplicity in algebra traces back to the work of Samuel in 1951 in connections with intersection theory of algebraic varieties. Multiplicities are closely related to the theory of convex bodies, and this relation is an active research topic lying in the interaction of Commutative Algebra, Combinatorics, and Algebraic Geometry. In this talk we will discuss some of the history of this topic and its applications. We also report on recent results in joint work with  Federico Castillo, Yairon Cid-Ruiz, Binglin Li, Fatemeh Mohammadi, and Naizhen Zhang. In one of these results, we show that mixed volumes of arbitrary convex bodies can be interpreted algebraically as mixed multiplicities of graded families of monomial ideals.  In another result, we present a complete characterization of the positivity of multidegrees of multiprojective algebraic varieties and establish a combinatorial description using convex geometry. We also use our methods to prove that double Schubert polynomials have Saturated Newton Polytopes, which settles a conjecture by Monical, Tokcan and Yong.