Alexandrov-Fenchel inequality in geometry and algebra

This is the first part of two pre-talks for June Huh's talks on Thursday and Friday. We begin by discussing the famous Brunn-Minkowski inequality and its stronger form Alexandrov-Fenchel inequality for volumes of convex bodies in the Euclidean space. We then discuss how exactly the same inequalities are satisfied between number of solutions of systems of algebraic equations. Some of the breakthrough results of June Huh relate to applying and/or proving versions of these inequalities in combinatorial contexts resolving several long-standing conjectures in combinatorics.