Mon, July 13, 3pm ET (8pm BST, 7am Tue NZST)
Federico Ardila, SFSU, U. de Los Andes
Geometry of Matroids
Zoom [email rose.mccarty ~at~ uwaterloo ~.~ ca if you need the password]

Abstract:
Matroid theory had its origins in linear algebra and graph theory, and it turns out to have deep connections with many other fields. With time, the geometric roots of the field have grown much deeper, bearing many new fruits. The interplay between matroid theory and algebraic geometry has recently led to the solution of long-standing combinatorial questions. Perhaps more importantly, it has opened up new and interesting research directions at the intersection of combinatorics, algebra, and geometry.

This talk will discuss my recent joint work with Graham Denham and June Huh, where we use ideas from Lagrangian geometry to prove Brylawski and Dawson’s conjectures on the log-concavity of the h-vector of a matroid. I will gear the talk towards a combinatorial audience, and assume no previous knowledge of algebraic geometry.

Mon, July 6, 3pm ET (8pm BST, 7am Tue NZST)
Marthe Bonamy, Univ. Bordeaux, CNRS
Graph classes and their Asymptotic Dimension
Zoom [email rose.mccarty ~at~ uwaterloo ~.~ ca if you need the password]

Abstract:
Introduced in 1993 by Gromov in the context of geometric group theory, the asymptotic dimension of a graph class measures how much “contact” is necessary between balls of “bounded” diameter covering a graph in that class. This concept has connections with clustered coloring or weak diameter network decompositions. While it seems surprisingly fundamental, much remains unknown about this parameter and it displays intriguing behaviours. We will provide a gentle exposition to the area: from the state of the art to the main tools and questions, including some answers.

This is joint work with Nicolas Bousquet, Louis Esperet, Carla Groenland, François Pirot and Alex Scott.

Hello everyone.

As you have likely already noticed, there is no talk in the seminar series today. As several other seminars have paused talks over these summer months, we are expanding the focus of the series to “Graphs and Matroids”.

We’re excited to have Marthe Bonamy giving the first talk next Monday under the new name, and Federico Ardila speaking the following week. As usual, more details will follow. See you next Monday!

Mon, June 22, 3pm ET (8pm BST, 7am Tue NZST)
Matthew Kwan, Stanford University
Halfway to Rota’s basis conjecture
Zoom [email rose.mccarty ~at~ uwaterloo ~.~ ca if you need the password]

Abstract:
In 1989, Rota made the following conjecture. Given $n$ bases $B_1,\ldots,B_n$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_i$ (we call such bases transversal bases). Rota’s basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers. In this talk we introduce the conjecture and its generalisation to matroids, and we outline a proof of the result that one can always find $(1/2−o(1))n$ disjoint transversal bases (improving on the previous record of $\Omega(n/\log{n})$). This talk will be accessible to non-matroid theorists.

Joint work with Matija Bucic, Alexey Pokrovskiy, and Benny Sudakov.