Difference between revisions of "Matroids seminar"

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<div><i>The multivariate Tutte polynomial of a flag matroid</i></div>
 
<div><i>The multivariate Tutte polynomial of a flag matroid</i></div>
 
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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.
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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 [https://arxiv.org/abs/1902.03719 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.
 
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Revision as of 22:59, 24 February 2019

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

1/18/2019
Introduction to matroids

We'll cover the basic definitions and some examples, roughly following these notes.

1/25/2019 & 2/1/2019
Algebraic matroids in action

We discuss algebraic matroids and their applications; see Algebraic Matroids in Action.

2/8/2019
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

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.

2/22/2019
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.

3/1/2019
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.