The matroids seminar & reading group meets 10:00--10:45 on Fridays in Van Vleck 901 in order to discuss matroids from a variety of viewpoints. In particular, we aim to
- survey open conjectures and recent work in the area
- compute many interesting examples
- discover concrete applications
We are happy to have new leaders of the discussion, and the wide range of topics to which matroids are related mean that each week is a great chance for a new participant to drop in! If you would like to talk but need ideas, see the Matroids seminar/ideas page.
To help develop an inclusive environment, a subset of the organizers will be available before the talk in the ninth floor lounge to informally discuss background material e.g., "What is a variety?", "What is a circuit?", "What is a greedy algorithm?" (this is especially for those coming from an outside area).
Organizers: Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez
Jose Israel Rodriguez
Planar pentads, polynomial systems, and polymatroids
Computing exceptional sets using fiber products naturally yields multihomogeneous systems of polynomial equations. In this talk, I will utilize a variety of tools from the forthcoming paper "A numerical toolkit for multiprojective varieties" to work out an example from kinematics: exceptional planar pentads. In particular, we will derive a multihomogeneous polynomial system whose solutions have meaning in kinematics and discuss how polymatroids play a role in describing the solutions.
Binary matroids and Seymour's decomposition in coding theory
We will begin by discussing the equivalence between a binary matroid and a binary linear code. And then following this paper and this one, we will describe the Maximum Likelihood decoding problem and then outline how Seymour's decomposition theorem for regular matroids led to a polynomial time algorithm on a subclass of binary linear codes.
The geometry of thin Schubert cells
We will cover the distinction between the thin Schubert cell of a matroid and the realization space of a matroid, how to compute examples, Mnev universality, and time permitting, maps between thin Schubert cells.
Plague and pestilence!
The multivariate Tutte polynomial of a flag matroid
Flag matroids are combinatorial objects whose relation to ordinary matroids are akin to that of flag varieties to Grassmannians. We define a multivariate Tutte polynomial of a flag matroid, and show that it is Lorentzian in the sense of Branden-Huh '19. As a consequence, we obtain a flag matroid generalization of Mason’s conjecture concerning the f-vector of independent subsets of a matroid. This is an on-going joint work with June Huh.
We outline the original formulation of matroid polytopes as moment polytopes of subvarieties of the Grassmanian, following Combinatorial Geometries, Convex Polyhedra, and Schbert Cells.
Proving the Heron-Rota-Welsh conjecture
We outline the proof of the Heron-Rota-Welsh conjecture given by Adiprasito, Huh, and Katz in their paper Hodge theory for combinatorial geometries
|1/25/2019 & 2/1/2019||
Algebraic matroids in action
We discuss algebraic matroids and their applications; see Algebraic Matroids in Action.
Introduction to matroids
We'll cover the basic definitions and some examples, roughly following these notes.