Combinatorial Algebraic Geometry: Mixed Volumes of Normal Complexes

The establishment of the log-concavity of characteristic polynomials of matroids was a result that took decades to confirm and connected ideas from combinatorics, algebra, and algebraic geometry.  In this talk, we will be exploring how this result can be achieved from the perspective of volume theory and polyhedral geometry.  In particular, we will explore the conditions under which the mixed volumes of a normal complex - an orthogonal truncation of a polyhedral fan - satisfy the Alexandrov-Fenchel inequalities. By then specializing to Bergman fans of matroids, we give a new proof of the log-concavity of characteristic polynomials of matroids.