YouTube recording: https://www.youtube.com/watch?v=2ga1XlPid9c

Time: Thursday, Mar 16, 3pm ET (beware of U.S./Canada time change on Mar 12!)
Zoom: https://uwaterloo.zoom.us/j/94977150195
pwd=LzRHN3NVcWtGbWlOVGNmakJQenF1Zz09
Passcode: kroeker1

Speaker: Matthew Kroeker, University of Waterloo
Title: Unavoidable Flats in Matroids Representable over a Prime Field

Abstract: The Sylvester-Gallai Theorem says that every rank-3 real-representable matroid contains a two-point line. A high-dimensional generalization of this result, due to Hansen, implies that every simple real-representable matroid of sufficiently high rank contains a rank-k independent flat. One only needs to look as far as the binary projective plane to see that such a result cannot hold for the class of matroids representable over a finite field. In light of this, we ask whether it is possible to determine a small list of “unavoidable” rank-k flats guaranteed to exist in a simple GF(q)-representable matroid of sufficiently high rank. We will answer this question for the case of prime fields: in particular, we show that, for any prime p and positive integer k, any simple GF(p)-representable matroid of sufficiently high rank contains a rank-k flat which is either independent, or is a projective or affine geometry. This is joint work with Jim Geelen.

YouTube recording: https://www.youtube.com/watch?v=veVUV1Ti0Wc

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

Speaker: Raphael Steiner, ETH Zürich
Title: Coloring hypergraphs with excluded minors

Abstract: Hadwiger’s conjecture, among the most famous open problems in graph theory, states that every graph that does not contain $K_t$ as a minor is properly (t−1)-colorable. The purpose of this work is to demonstrate that a natural extension of Hadwiger’s problem to hypergraph coloring exists, and to derive some first partial results and applications. Generalizing ordinary graph minors to hypergraphs, we say that a hypergraph $H_1$ is a minor of a hypergraph $H_2$, if a hypergraph isomorphic to $H_1$ can be obtained from $H_2$ via a finite sequence of vertex- and hyperedge-deletions, and hyperedge contractions. We first show that a weak extension of Hadwiger’s conjecture to hypergraphs holds true: For every positive integer t, there exists a finite (smallest) integer h(t) such that every hypergraph with no $K_t$-minor is h(t)-colorable, and we prove $$\lceil 3/2(t-1) \rceil \le h(t) \le 2g(t)$$ where g(t) denotes the maximum chromatic number of graphs with no $K_t$-minor. Using the recent result by Delcourt and Postle that $g(t) = O(t \log\log t)$, this yields $h(t) = O(t\log\log t)$. We further conjecture that $h(t) = \lceil 3/2(t-1) \rceil$, i.e., that every hypergraph with no $K_t$-minor is $\lceil 3/2(t-1) \rceil$-colorable for all t, and prove this conjecture for all hypergraphs with independence number at most 2. By considering special classes of hypergraphs, the above additionally has some interesting applications for ordinary graph coloring, such as:
-graphs of chromatic number $Ckt\log\log t$ contain $K_t$-minors with k-edge-connected branch-sets,
-graphs of chromatic number $Cqt\log\log t$ contain $K_t$-minors with modulo-q-connected branch sets,
-by considering cycle hypergraphs of digraphs we recover known results on strong minors in digraphs of large dichromatic number as special cases.

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.