Geometry and Physics Seminar: Jeffrey Giansiracusa

In the standard approach to tropicalization, an algebraic subset X of a toric variety over a non-archimedean valued field k is sent to a weighted polyhedral set Trop(X) which we think of as a combinatorial shadow of X. The result depends only on the k-points of X. A system of polynomial equations often contains more information than the set of its solutions over a field, and the philosophy of scheme theory is that we should treat the system of equations itself as a fundamental geometric object from which the solution set is derived. Scheme-theoretic tropicalization is about realizing Trop(X) as the solution set to an underlying system of polynomial equations over the idempotent semiring of tropical numbers - a system that is constructed in a canonical way from the equations defining X. The theory involves the field with one element, and with these ideas the Berkovich analytification appears as the universal tropicalization of X and as the moduli space of valuations on X.