EVENT DETAILS AND ABSTRACT

Title: Matroids and Grassmannians over Hyperfields
Speaker: Matthew Baker
Speaker Info: Georgia Tech
Brief Description:
Special Note:
Abstract:

A hyperfield is an algebraic structure satisfying similar axioms to a field, except that addition is allowed to be multi-valued. The simplest example of a hyperfield which is not a field is the Krasner hyperfield, which consists of the elements {0,1} with the usual multiplication, but with the strange-looking hyperaddition law 1+1 = {0,1}. Given a hyperfield F and a positive integer n, we will give several equivalent definitions for what it means for a subset V of F^n to be a linear subspace. When F is a field, this coincides with the usual notion from linear algebra, but when F is the Krasner hyperfield, for example, it corresponds to the combinatorial notion of a matroid. For other choices of F, we recover the notions of oriented matroids and tropical linear space. The collection of linear subspaces of F^n of some fixed rank r is parametrized by a projective F-variety which generalizes the usual Grassmannian. This is joint work with Nathan Bowler.