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.