Isoperimetric type inequalities in geometry and algebra

Speaker: Professor Askold Khovanskii 
Affiliation: University of Toronto

Abstract

A beautiful isoperimetric inequality was known to the Ancient Greeks claiming that the circle has the largest area amongst all planar domains with a fixed boundary length. This inequality has many interesting geometric generalizations. One of them is the famous Brunn-Minkowsky inequality which relates volumes of convex bodies and the volume of their Minkowsky sum. One can define the mixed volume of n-convex bodies in n-dimensional real vector space. Another one, the more general Alexandrov-Fenchel inequality relates certain mixed volumes of dierent n-tuples of convex bodies. Quite surprisingly, there are analogous inequalities in algebra. First of all, the BKK (Bernstein-Khovanskii-Koushnirenko) theorem computes the number of solutions of n generic polynomial equations with given Newton polyhedra in (C*)n in terms of the mixed volume of their Newton polyhedra. Indeed, such inequalities, known as Khovanskii-Teissier inequality, were eventually found. Furthermore, a group of mathematicians, in connection with Field's Laureate June Huh, has recently discovered multiple fascinating applications for these inequalities.

In this talk, I will discuss such inequalities and other relations between Algebraic and Convex Geometries. The talk will be accessible to a broad mathematical audience.