What is the tropicalization 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 on how to extract tropical information from a matrix over a non-Archimedean field. Each perspective will give rise to inherently quite different phenomena. Central instances of this rich panorama include the tropical geometry of vector bundles, logarithmic concavity results for valuated (bi-)matroids (using techniques from combinatorial Hodge theory), and the geometry of affine buildings.

This talk draws from joint work with Andreas Gross and Dmitry Zakharov; Andreas Gross, Inder Kaur, and Annette Werner; Felix Röhrle; Jeff Giansiracusa, Felipe Rincon, and Victoria Schleis; Luca Battistella, Kevin Kühn, Arne Kuhrs, and Alejandro Vargas; as well as with Desmond Coles.