# Matroids seminar: Difference between revisions

No edit summary |
No edit summary |
||

(39 intermediate revisions by 3 users not shown) | |||

Line 1: | Line 1: | ||

The matroids seminar & reading group meets '''10 | The matroids seminar & reading group meets '''9:10--10:50 on Fridays on Zoom''' in order to discuss matroids from a variety of viewpoints. | ||

In particular, we aim to | In particular, we aim to | ||

* survey open conjectures and recent work in the area | * survey open conjectures and recent work in the area | ||

* compute many interesting examples | * compute many interesting examples | ||

* discover concrete applications | * 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! | 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 | |||

<div style="font-weight:bold;">[https://www.math.wisc.edu/wiki/index.php/Applied_Algebra_Seminar_Spring_2020]</div>. | |||

|- style=" | |||

'''Organizers''': Colin Crowley, Connor Simpson; Jose Israel Rodriguez, Botong Wang | |||

{|cellpadding="8px" style="vertical-align:top; text-align:left; cellpadding:10px;" | |||

|- | |||

|09/25/2020 | |||

|We will discuss "Rigidity of linearly constrained frameworks in d-dimensions" based on the work of Tony Nixon (Lancaster University, https://www.lancaster.ac.uk/maths/people/anthony-nixon). See a virtual talk at 9am 9/24 at this link https://dibernstein.github.io/VirtualSeminar.html or check out this paper https://arxiv.org/abs/2005.11051 on the topic. | |||

|- | |||

|} | |||

'''Past Seminars''' (Organized by Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez) | |||

{|cellpadding="8px" style="vertical-align:top; text-align:left; cellpadding:10px;" | |||

|- | |||

|5/03/2019 | |||

| | |||

<div style="font-weight:bold;">[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]</div> | |||

<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"> | |||

<div><i>A flip-free proof of the Heron-Rota-Welsh conjecture</i></div> | |||

<div class="mw-collapsible-content"> | |||

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. | |||

</div></div> | |||

|- | |||

|4/12/2019 & 4/19/2019 | |||

| | |||

<div style="font-weight:bold;">No seminar</div> | |||

<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"> | |||

<div><i>Many organizers are traveling.</i></div> | |||

<div class="mw-collapsible-content"> | |||

None. | |||

</div></div> | |||

|- | |||

|4/05/2019 | |||

| | |||

<div style="font-weight:bold;">Jose Israel Rodriguez</div> | |||

<div class="mw-collapsible mw-collapsed" data-expandtext="Show abstract" data-collapsetext="Hide abstract" style="width:450px; overflow:auto;"> | |||

<div><i>Planar pentads, polynomial systems, and polymatroids</i></div> | |||

<div class="mw-collapsible-content"> | |||

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. | |||

</div></div> | |||

|- | |||

|3/29/2019 | |||

| | |||

<div style="font-weight:bold;">[https://sites.google.com/view/colincrowley/home Colin Crowley]</div> | |||

<div><i>Binary matroids and Seymour's decomposition in coding theory</i></div> | |||

<div class="mw-collapsible-content"> | |||

We will begin by discussing the equivalence between a binary matroid and a binary linear code. And then following [https://ac.els-cdn.com/009589568990052X/1-s2.0-009589568990052X-main.pdf?_tid=a1d8598e-c5f2-4c07-8d7b-0843e88c416f&acdnat=1552505468_33915774714b2e407e4fd7001ba33100 this paper] and [https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=4544972 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. | |||

</div></div> | |||

|- | |||

|3/15/2019 | |||

| | |||

<div style="font-weight:bold;">[https://sites.google.com/site/dcorey2814/home Daniel Corey]</div> | |||

<div><i>The geometry of thin Schubert cells</i></div> | |||

<div class="mw-collapsible-content"> | |||

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. | |||

</div></div> | |||

|- | |||

|3/8/2019 | |||

| | |||

<div style="font-weight:bold;">Vladmir Sotirov</div> | |||

<div><i>is sick</i></div> | |||

<div class="mw-collapsible-content"> | |||

Plague and pestilence! | |||

</div></div> | |||

|- | |||

|3/1/2019 | |||

| | |||

<div style="font-weight:bold;">[https://math.berkeley.edu/~ceur/ Chris Eur]</div> | |||

<div><i>The multivariate Tutte polynomial of a flag matroid</i></div> | |||

<div class="mw-collapsible-content"> | |||

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. | |||

</div></div> | |||

|- | |||

|2/22/2019 | |||

| | |||

<div style="font-weight:bold;">[https://www.math.wisc.edu/~wang/ Botong Wang]</div> | |||

<div><i>The Kazhdan-Lusztig polynomial of a matroid</i></div> | |||

<div class="mw-collapsible-content"> | |||

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 [https://arxiv.org/pdf/1611.07474.pdf these] [https://arxiv.org/pdf/1412.7408.pdf two] papers. | |||

</div></div> | |||

|- | |||

|2/15/2019 | |||

| | |||

<div style="font-weight:bold;">[https://sites.google.com/view/colincrowley/home Colin Crowley]</div> | |||

<div><i>Matroid polytopes</i></div> | |||

<div class="mw-collapsible-content"> | |||

We outline the original formulation of matroid polytopes as moment polytopes of subvarieties of the Grassmanian, following [http://www.math.ias.edu/~goresky/pdf/combinatorial.jour.pdf Combinatorial Geometries, Convex Polyhedra, and Schbert Cells]. | |||

</div></div> | |||

|- | |||

|2/8/2019 | |||

| | |||

<div style="font-weight:bold;">[http://www.math.wisc.edu/~csimpson6/ Connor Simpson]</div> | |||

<div><i>Proving the Heron-Rota-Welsh conjecture</i></div> | |||

<div class="mw-collapsible-content"> | |||

We outline the proof of the Heron-Rota-Welsh conjecture given by Adiprasito, Huh, and Katz in their paper [https://arxiv.org/abs/1511.02888 Hodge theory for combinatorial geometries] | |||

</div></div> | |||

|- | |||

|1/25/2019 & 2/1/2019 | |||

| | |||

<div style="font-weight:bold;">[https://www.math.wisc.edu/~jose/ Jose Israel Rodriguez]</div> | |||

<div><i>Algebraic matroids in action</i></div> | |||

<div class="mw-collapsible-content"> | |||

We discuss algebraic matroids and their applications; see [https://arxiv.org/abs/1809.00865 Algebraic Matroids in Action]. | |||

</div></div> | |||

|- | |||

|1/18/2019 | |1/18/2019 | ||

| | | | ||

<div style="font-weight:bold;">Daniel Corey</div> | <div style="font-weight:bold;">[https://sites.google.com/site/dcorey2814/home Daniel Corey]</div> | ||

<div><i>Introduction to matroids</i></div> | <div><i>Introduction to matroids</i></div> |

## Latest revision as of 19:24, 21 September 2020

The matroids seminar & reading group meets **9:10--10:50 on Fridays 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

.

**Organizers**: Colin Crowley, Connor Simpson; Jose Israel Rodriguez, Botong Wang

09/25/2020 | We will discuss "Rigidity of linearly constrained frameworks in d-dimensions" based on the work of Tony Nixon (Lancaster University, https://www.lancaster.ac.uk/maths/people/anthony-nixon). See a virtual talk at 9am 9/24 at this link https://dibernstein.github.io/VirtualSeminar.html or check out this paper https://arxiv.org/abs/2005.11051 on the topic. |

**Past Seminars** (Organized by Colin Crowley, Connor Simpson; Daniel Corey, Jose Israel Rodriguez)

5/03/2019 |
A flip-free proof of the Heron-Rota-Welsh conjectureThe 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 polymatroidsComputing 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 |
Binary matroids and Seymour's decomposition in coding theoryWe 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 |
The geometry of thin Schubert cellsWe 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 sickPlague and pestilence! |

3/1/2019 |
The multivariate Tutte polynomial of a flag matroidFlag 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 | |

2/15/2019 |
Matroid polytopesWe 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 |
Proving the Heron-Rota-Welsh conjectureWe 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 actionWe discuss algebraic matroids and their applications; see Algebraic Matroids in Action. |

1/18/2019 |
Introduction to matroidsWe'll cover the basic definitions and some examples, roughly following these notes. |