For a given hypergraph $H$ and a vertex $v\in V(H)$, consider a random matching $M$ chosen uniformly from the set of all matchings in $H.$ In $1995,$ Kahn conjectured that if $H$ is a $d$-regular linear $k$-uniform hypergraph, the probability that $M$ does not cover $v$ is $(1 + o_d(1))d^{-1/k}$ for all vertices $v\in V(H)$. This conjecture was proved for $k = 2$ by Kahn and Kim in 1998. In this paper, we disprove this conjecture for all $k \geq 3.$ For infinitely many values of $d,$ we construct $d$-regular linear $k$-uniform hypergraph $H$ containing two vertices $v_1$ and $v_2$ such that $\mathcal{P}(v_1 \notin M) = 1 – \frac{(1 + o_d(1))}{d^{k-2}}$ and $\mathcal{P}(v_2 \notin M) = \frac{(1 + o_d(1))}{d+1}.$ The gap between $\mathcal{P}(v_1 \notin M)$ and $\mathcal{P}(v_2 \notin M)$ in this $H$ is best possible. In the course of proving this, we also prove a hypergraph analog of Godsil’s result on matching polynomials and paths in graphs, which is of independent interest.

In some sense, matroids are generalisations of graphs. The idea of graph minors extends to matroids, and so does the idea of a minor-closed class. We can think of a minor-closed class of matroids as being an analogue to the class of graphs embeddable on a surface. Any such class of graphs has a corresponding class of minimal forbidden minors, and these forbidden minors characterise the class. A minor-closed class of matroids is characterised by its minimal forbidden minors in the same way. Rota’s conjecture is the most famous problem in matroid theory. It says that when F is a finite field, there is a finite number of minimal forbidden minors for the class of matroids that can be represented by vectors over the field of scalars F. A proof has been announced by Geelen, Gerards, and Whittle. Gain-graphic matroids are analogues to matroids represented by vectors: instead of representing the matroid using numbers from a field, we use elements from a group. So we can ask for an analogue of Rota’s conjecture, except for gain-graphic matroids. In this talk I will outline our intended path towards Rota’s conjecture for gain-graphic matroids. This is joint work with Daryl Funk.

Tropical geometry replaces usual addition and multiplication with tropical addition (the min) and tropical multiplication (the sum), which offers a polyhedral interpretation of algebraic variety. This talk aims to pitch the usefulness of tropical geometry in understanding classical algebraic geometry. As an example, we introduce the tropicalization of the variety of symmetric rank 2 matrices. We discuss that this tropicalization has a simplicial complex structure as the space of symmetric bicolored trees. As a result, we show that this space is shellable and delve into its matroidal structure. It is based on the joint work with May Cai and Josephine Yu.