Matroids seminar

The matroids seminar & reading group meets 10:10--10:50 on Thursdays on Zoom 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

For updates, join our mailing list, matroids [at] lists.wisc.edu

During the Fall 2020 semester the seminar will be discussion based on a predetermined topic. See below.

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.

A related seminar is the Applied Algebra seminar

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Organizers: Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez

5/03/2019

Connor Simpson A flip-free proof of the Heron-Rota-Welsh conjecture The simplicial presentation of a matroid yields a flip-free proof of the Kahler package in degree 1 for the Chow ring of a matroid, which is enough to give a new proof of the Heron-Rota-Welsh conjecture. This talk is more or less a continuation of the one that Chris Eur gave earlier in the semester in the algebra seminar, and is based on the same joint work with Spencer Backman and Chris Eur. Past Seminars (Organized by Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez) 5/03/2019 Connor Simpson A flip-free proof of the Heron-Rota-Welsh conjecture The simplicial presentation of a matroid yields a flip-free proof of the Kahler package in degree 1 for the Chow ring of a matroid, which is enough to give a new proof of the Heron-Rota-Welsh conjecture. This talk is more or less a continuation of the one that Chris Eur gave earlier in the semester in the algebra seminar, and is based on the same joint work with Spencer Backman and Chris Eur. 4/12/2019 & 4/19/2019 No seminar Many organizers are traveling. None. 4/05/2019 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. 3/29/2019 Colin Crowley 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. 3/15/2019 Daniel Corey 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. 3/8/2019 Vladmir Sotirov is sick Plague and pestilence! 3/1/2019 Chris Eur 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. 2/22/2019 Botong Wang The Kazhdan-Lusztig polynomial of a matroid Classically, Kazdhan-Lusztig polynomials are associated to intervals of the Bruhat poset of a Coxeter group. We will discuss an analogue of Kazdhan-Lusztig polynomials for matroids, including results and conjectures from these two papers. 2/15/2019 Colin Crowley Matroid polytopes We outline the original formulation of matroid polytopes as moment polytopes of subvarieties of the Grassmanian, following Combinatorial Geometries, Convex Polyhedra, and Schbert Cells. 2/8/2019 Connor Simpson 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 Jose Israel Rodriguez Algebraic matroids in action We discuss algebraic matroids and their applications; see Algebraic Matroids in Action. 1/18/2019 Daniel Corey Introduction to matroids We'll cover the basic definitions and some examples, roughly following these notes.