Back to Commutative Algebra Seminar: Fall 2022
Commutative Algebra Seminar
Date: Wednesday, Aug 31, 2022
Time: 2:00PM - 3:00PM
Location: LCB 222
Matt Baker
Georgia Tech
Title |
Non-archimedean and tropical geometry, algebraic groups, moduli spaces of matroids, and the field with one element |
Abstract |
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. |
Commutative Algebra