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
Youtube
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.
Mon, June 15, 3pm EST (8pm BST, 7am Tue NZST)
Rutger Campbell, University of Waterloo
Matroids doing algebra
Youtube
Abstract:
We will look at matroids that enforce algebraic relations in their representations. I will briefly survey matroidal results that are then attained from basic algebraic conditions.
Mon, June 8, 3pm EST (8pm BST, 7am Tue NZST)
Relinde Jurrius, Netherlands Defence Academy
q-Analogues in combinatorics
Youtube
Abstract:
A q-analogue is, roughly speaking, what happens if we generalise from finite sets to finite dimensional vector spaces. The main focus of this talk will be, of course, the q-analogue of a matroid. Sometimes the change from sets to spaces goes very smoothly, but sometimes strange things (appear to) happen. This will be illustrated by discussing several cryptomorphic definitions of q-matroids. As an application of q-matroids, we link them to subspace designs, the q-analogue of combinatorial designs.