What is the combinatorial shadow of a matrix?

Tropicalization is a process that associates to an algebro-geometric object a piecewise linear polyhedral shadow that captures its essential combinatorial structure. In this talk, I will give an overview of the numerous ways of how to extract tropical information from a matrix. Our focus will be on naturally occurring logarithmically concave sequences associated to a matrix (or more generally a matroid respectively a bimatroid), an area of study, which in recent years has tremendous progress due to the introduction of methods originating from Hodge theory. A particular emphasis will be made to make this story accessible; a background in algebraic geometry is not necessary to follow this talk.

This talk draws from joint work with Felix Röhrle as well as with Jeff Giansiracusa, Felipe Rincon, and Victoria Schleis.