YouTube recording: https://www.youtube.com/watch?v=6hmhSYado9Q

Time: Thursday, Feb 16, 3pm EST (8pm GMT, 9am Fri NZDT)
Zoom: https://gatech.zoom.us/j/8802082683

Speaker: Relinde Jurrius, Netherlands Defense Academy
Title: The combinatorial derived matroid
Abstract: The question of “dependencies between dependencies” in matroids has been raised in the 1960’s by Rota and Crapo, in various formulations (here is a nice overview elsewhere on this blog: matroidunion.org/?p=2628). For a matroid $M$, the derived matroid is a matroid that has as ground set the set of (co)circuits of $M$. Over time, various definitions of a derived matroid have been proposed. However, none of them completely solved the question of Rota and Crapo: these definitions depend on a representation, for example, or are not unique, or don’t always exist.
In a recent paper with Olga Kuznetsova and Ragnar Freij-Hollanti, we proposed a definition of a derived matroid that is purely combinatorial. The combinatorial derived matroid is defined via the rank function of $M$, and via an operation that resembles a closure operation on the collection of dependent sets. It is therefore uniquely defined for any matroid. In this talk I will motivate this definition and discuss some examples and desirable properties of this definition. Time permitting we will discuss open questions and links to previous definitions.

YouTube recording: https://youtu.be/UdjhEgSKQus

Time: Thursday, Feb 2, 3pm EST (8pm GMT, 9am Fri NZDT)
Zoom: https://gatech.zoom.us/j/8802082683

Speaker: Tom Zaslavsky, Binghamton University
Title: Matroids of gain signed graphs
Abstract: For standard affinographic hyperplane arrangements (a.k.a. deformations of
the Type A root system arrangement or “braid” arrangement), integral gain
graphs give a simpler method to compute the characteristic polynomial, a
fundamental invariant.  For more general affinographic arrangements
(a.k.a. deformations of the Type B root system arrangement), one has to
combine gains with signs.  How to do this has been a puzzle.  The obvious
method is to put signs on top of gains.  The right method is to put gains
on top of signs.  Laura Anderson, Ting Su, and I found out how to do this,
constructing the natural matroid and the corresponding semimatroid, which
latter gives the characteristic polynomial of these more general
arrangements when the gain group is the additive group of integers.  I
will explain some of this.  It does get complicated.

YouTube recording: https://youtu.be/acJnDJ7sZPM

The online Graphs and Matroids seminar will meet over Zoom every other Thursday, starting on Jan 19 with a talk by Johannes Carmesin. We hope to see you there!

Time: Thursday, Jan 19, 3pm EST (8pm GMT, 9am Fri NZDT)
Zoom: https://gatech.zoom.us/j/8802082683

Speaker: Johannes Carmesin, University of Birmingham
Title: Dual matroids of 2-complexes
Abstract: A fundamental theorem in matroid theory is Whitney’s theorem,
saying that a graph is planar if and only if the dual of its cycle matroid
is graphic. Like plane graphs, 2-complexes C embedded in 3-space have dual
graphs; their vertices are the chambers of the embedding and two chambers
are adjacent if they share a face. This dual graph of C can be described
through a dual matroid of the 2-complex C. How can Whitney’s theorem be
extended to 2-complexes?