Learning Seminar in Algebraic Combinatorics @ University of Michigan

In a previous lecture, we learnt about tropical linear spaces in terms of valuated matroids, or tropical Plucker vectors. In this talk, I'll give a quick overview of some other ways to think about tropical linear spaces: matroid decompositions of the hypersimplex, Chow quotients of the Grassmannian, and membranes. I'll focus on the last of these. A result of Keel and Tevelev states that tropical linear spaces can be identified with membranes, certain subcomplexes of a (Bruhat-Tits) affine building.

Building off of last week's talk, we will discuss the tropical totally positive Grassmannian, Trop+(Gr(k,n)). First introduced by Speyer and Williams in 2003, the tropical totally positive Grassmannian came out of work studying the totally positive part of the tropicalization of an affine variety. After briefly looking at those results, we'll focus our attention on the fan structure of the tropical totally positive Grassmannian. If there's time, we'll show that the fan associated to Trop+(Gr(2,n)) is combinatorially the fan dual to the type A_{n-3} associahedra.

Matroids are hard. Valuated matroids generalize matroids. Tropical linear spaces are equivalent to valuated matroids. Therefore, tropical linear spaces are hard, so are their parameter spaces: the Dressians, the tropical analog of the Grassmannians. Inside the Dressian there is a nicer subset, called the tropical Grassmannian, that parametrizes the tropicalized linear spaces. I will familiarize the audience with these concepts with a lot of examples.

We introduce the notion of a balanced weighted polyhedral complex, and prove the structure theorem for tropical varieties. This says that the tropicalization of an irreducible d-dimensional variety is a balanced weighted rational polyhedral complex which is pure of dimension d and connected through codimension 1.

In this talk, we introduce the tropical hypersurface associated to a multivariate Laurent polynomial, as well as more general tropical varieties arising from classical subvarieties of the torus. We will discuss Kapranov's theorem and present the generalized Fundamental Theorem of Tropical Algebraic Geometry, both of which unite classical algebro-geometric objects to their tropical counterparts.

In this talk I will introduce Gröbner basis in a field with valuation and the construction of Gröbner complexes, which will be the ambient space for tropical varieties. Relevant background on valuations and polyhedral geometry will also be covered in the talk.

I will give an introduction to the subject and go over some basics of tropical geometry, following chapter one of Maclagan-Sturmfels' book "Introduction to Tropical Geometry".