#### Hall 123

#### Non-Archimedean Combinatorics

#### Maria Angelica Cueto

##### Ohio State University

*Non-Archimedean analytic geometry, as developed by Berkovich, is*

a variation of classical complex analytic geometry for non-Archimedean

fields such as p-adic numbers. Solutions to a system of polynomial equations

over these fields form a totally disconnected space in their natural

topology. The process of analytification adds just enough points to make

them locally connected and Hausdorff. The resulting spaces are technically

difficult to study but, notably, their heart is

combinatorial: they can be examined through the lens of tropical and

polyhedral geometry.

I will illustrate this powerful philosophy through concrete examples,

including elliptic curves, the tropical Grassmannian of planes of

Speyer-Sturmfels, and a compactification of the well-known space of

phylogenetic trees of Billera-Holmes-Vogtmann.

Done