A Latin square of order n is an n by n grid filled with n different symbols so that every symbol occurs exactly once in each row and each column, while a transversal in a Latin square is a collection of cells which share no row, column or symbol. The Ryser-Brualdi-Stein conjecture states that every Latin square of order n should have a transversal with n-1 elements, and one with n elements if n is odd. In 2020, Keevash, Pokrovskiy, Sudakov and Yepremyan improved the long-standing best known bounds on this conjecture by showing that a transversal with n-O(log n/loglog n) elements exists in any Latin square of order n. In this talk, I will discuss how to show, for large n, that a transversal with n-1 elements always exists.

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Will balloons reach the origin infinitely often or not? We answer this question for various underlying spaces. En route we find a new(ish) 0-1 law, and generalize bounds on independent sets that are factors of IID on trees. Joint work with Omer Angel and Gourab Ray.

In 1983 Thomassen conjectured that every graph of sufficiently large average degree has a subgraph of large girth and large average degree. While this conjecture remains open, recent evidence suggests that something stronger might be true; perhaps the subgraph can be made induced when a clique and biclique are forbidden. We overview our proof for removing 4-cycles from $K_{t,t}$-free bipartite graphs. Moreover, we discuss consequences to tau-boundedness, which is an analog of chi-boundedness.

A 1-independent bond percolation model on a graph $G$ is a probability distribution on the spanning subgraphs of $G$ in which, for all vertex-disjoint sets of edges $S_1$ and $S_2$, the states (i.e. present or not present) of the edges in $S_1$ are independent of the states of the edges in $S_2$. Such models typically arise in renormalisation arguments applied to independent percolation models, or percolation models with finite range dependencies. A 1-independent model is said to percolate if the random subgraph has an infinite component with positive probability. In 2012 Balister and Bollobás defined $p_{\textrm{max}}(G)$ to be the supremum of those $p$ for which there exists a 1-independent bond percolation model on $G$ in which each edge is present in the random subgraph with probability at least $p$ but which does not percolate. A fundamental and challenging problem in this area is to determine, or give good bounds on, the value of $p_{\textrm{max}}(G)$ when $G$ is the lattice graph $\mathbb{Z}^2$. Since $p_{\textrm{max}}(\mathbb{Z}^n)\leq p_{\textrm{max}}(\mathbb{Z}^{n-1})$, it is also of interest to establish the value of $\lim_{n\to\infty}p_{\textrm{max}}(\mathbb{Z}^n)$.

In this talk we will present a significantly improved upper bound for this limit as well as improved upper and lower bounds for $p_{\textrm{max}}(\mathbb{Z}^2)$. We will also show that with high confidence we have $p_{\textrm{max}}(\mathbb{Z}^n)<p_{\textrm{max}}(\mathbb{Z}^2)$ for large $n$ and discuss some open problems concerning 1-independent models on other graphs.

This is joint work with Tom Johnston and Alex Scott.

This talk is motivated by work on the existence theory for viscoelasticity of Kelvin-Voigt type with non-convex stored energies (joint with K. Koumatos (U. of Sussex), C. Lattanzio and S. Spirito (U. of LAquila)), which shows propagation of H1-regularity for the deformation gradient of weak solutions for semiconvex stored energies. It turns out that weak solutions with deformation gradient in H1 are in fact unique in two-space dimensions, providing a striking analogy to corresponding results in the theory of 2D Euler equations with bounded vorticity.

While weak solutions still exist for initial data in L2, oscillations on the deformation gradi- ent can now persist and propagate in time. This can be seen via a counterexample indicating that for non-monotone stress-strain relations in 1-d oscillations of the strain lead to solutions with sustained oscillations. The existence of sustained oscillations in hyperbolic-parabolic system is then studied in several examples motivated by viscoelasticity and thermoviscoelas- ticity. Sufficient conditions for persistent oscillations are developed for linear problems, and examples in some nonlinear systems of interest. In several space dimensions oscillatory exam- ples are associated with lack of rank-one convexity of the stored energy. Nonlinear examples in models with thermal effects are also developed.