Abstract:

Log-concavity and unimodality are properties of a sequence of numbers. They often appear in mathematics, in particular combinatorics. Sometimes we can prove these properties using algebraic geometry. In this talk, I would like to explain recent results on log-concavity (and unimodality) for matroids, which are combinatorial generalizations of graphs and (linear independences of rows of) matrices. When matroids come from graphs or matrices, we can use algebraic geometry. In general, instead of it, we can use tropical geometry, a combinatorial analog of algebraic geometry.