YouTube recording: https://www.youtube.com/watch?v=OaRpqCco8ko

Time: Thursday, Apr 13, 3pm ET
Zoom: https://gatech.zoom.us/j/8802082683

Speaker: Kristyna Pekárková, Masaryk University
Title: Matroid-based approach to matrix sparsification

Abstract: Integer programming (IP) is a fundamental problem of discrete optimization;
its notable applications can be found, for example, in scheduling and planning,
string algorithms, or computational social choice. Solving instances of integer
programming is, in general, NP-hard, and a long line of research has been
devoted to identifying instances of IP that can be solved efficiently. In the
talk, we will explain how structural properties of matroids can be applied in
the context of integer programming. Our main focus is on matroid depth
parameters, namely contraction*-depth and deletion-depth. In particular, we
will discuss how these parameters can be used to design algorithms that
efficiently transform the constraint matrix of an instance of integer
programming into an equivalent sparse (and so computationally tractable)
form.

YouTube recording: https://www.youtube.com/watch?v=iIcl15EgNe4

Time: Thursday, Mar 30, 3pm ET
Zoom: https://gatech.zoom.us/j/8802082683

Speaker: Ruth Luo, University of South Carolina
Title: A hypergraph analog of Dirac’s Theorem for 2-connected graphs

Abstract: Every graph with minimum degree $k \geq 2$ contains a cycle of length at least $k+1.$ Dirac proved that if the graph is also 2-connected then in fact we can find a cycle of length at least $min\{2k, n\}$. We prove a hypergraph version of this theorem: a minimum degree condition that forces the existence of long Berge cycles in 2-connected, uniform hypergraphs. This is joint work with Alexandr Kostochka and Grace McCourt.

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.