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.