Number Theory Seminar: Non-archimedean and tropical geometry, algebraic groups, moduli spaces of matroids, and the field with one element

SEMINARS, NUMBER THEORY

Speaker:

Matt Baker - Georgia Teach

I will give an introduction to Oliver Lorscheid’s theory of ordered blueprints – one of the more successful approaches to “the field of one element” – and sketch its relationship to Berkovich spaces, tropical geometry, Tits models for algebraic groups, and moduli spaces of matroids. The basic idea for the latter two applications is quite simple: given a scheme over Z defined by equations with coefficients in {0,1,-1}, there is a corresponding “blue model” whose K-points (where K is the Krasner hyperfield) sometimes correspond to interesting combinatorial structures. For example, taking K-points of a suitable blue model for a split reductive group scheme G over Z gives the Weyl group of G, and taking K-points of a suitable blue model for the Grassmannian G(r,n) gives the set of matroids of rank r on {1,…,n}. Similarly, the Berkovich analytification of a scheme X over a valued field K coincides, as a topological space, with the set of T-points of X, considered as an ordered blue scheme over K. Here T is the tropical hyperfield, and T-points are defined using the observation that a (height 1) valuation on K is nothing other than a homomorphism to T.