# Learning Seminar in Algebraic Combinatorics @ University of Michigan

Topic: Matroids (Fall 2024)Meetings: Wednesdays 3pm--4pm in East Hall 4088

Organizers: Thomas Lam, George H. Seelinger, David E Speyer, and Lei Xue.

Previous year's talks

# Talks

## Abstract

I’ll provide an overview of (1) hyperplane arrangements (2) counting points on hyperplane arrangements (3) Rota’s conjecture and the idea of Huh’s proof of Rota’s conjecture (4) the connections to tropical geometry and (5) the recent idea of the “Chow ring of a matroid”. The point is to motivate you to be interested in these concepts, which we’ll be exploring throughout the term; many ideas will just be sketched.

## Abstract

As David Speyer discussed last week, hyperplane arrangements were a major motivation for the study of matroids. In this talk, we’ll take a more in-depth look at hyperplane arrangements with a focus on their characteristic polynomials. While initially defined in terms the point count over F_q, the characteristic polynomial can be computed entirely combinatorially from the intersection poset. Moreover, in certain situations, the characteristic polynomial has a physical interpretation; over R, it gives the number of connected components and, over C, it gives the Betti numbers. This talk will highlight these results and some of the techniques used to prove them.

## Abstract

In this talk, we will discuss the definition of matroids and their properties. As alluded to in previous talks, matroids generalize the notion of linear independence of subsets of some ambient set. We will discuss some examples of matroids and recipes to get them from more concrete objects. Furthermore, we will define terms such as basis, spanning sets, rank, and flats, as well as the associated lattice of flats of a matroid, generalizing ideas like the intersection poset from the theory of hyperplane arrangements.

## Abstract

In this talk, we will continue our introduction to matroids. We will discuss dual matroids, restriction and contraction, and realizability of matroids.

## Abstract

I will talk about the matroid polytope, their definition, characterization of matroids in terms of matroid polytopes, and more.

## Abstract

In this talk we first introduce another way of constructing new matroids from a given one M, by minimizing the linear functional \(w \cdot x\) over \(x\) in the matroid polytope of \(M\), for a fixed \(w \in \mathbb{R}^n\). We then use this to define the Bergman fan of a given matroid, which are spaces of weights \(w\) whose corresponding matroids do not contain loops. To understand the topology of the Bergman fan, we introduce a combinatorial gadget called phylogenetic trees which can be used to parametrize the Bergman fan, and give a nice subdivision of it.

## Abstract

I’ll start with Bergman’s 1971 definition of the logarithmic limit set of an ideal, and explain why we now call it the “tropicalization” of the ideal. I’ll then explain how, when we study tropicalizations of linear spaces, we get the Bergmann complex of a matroid, which Yucong spoke about last week. Finally, I’ll try to motivate the wonderful compactification of a hyperplane arrangement complement. Expect a fairly informal, loosely prepared, talk.

## Abstract

To prove log convexity of certain sequences of numbers, one wants to realize those numbers as intersection numbers of certain cohomology classes on some varieties. We will introduce one kind of such varieties called wonderful compactifications.

This construction compactifies the complement of hyperplane arrangements with nice boundary divisors. We will discuss two ways of constructing them, via ‘manually adding normal directions’ and via blow-up.