Georgia Institute of Technology College of Sciences

Langhede and Thomassen conjectured in 2020 that there exists a positive constant c such that every planar graph G with 5-correspondence assignment (L,M) has at least 2^{c v(G)} distinct (L,M)-colourings. I will discuss a proof of this conjecture (which relies on the hyperbolicity of a certain family of graphs), a generalization of this result to some other embedded graphs (again, relying on a hyperbolicity theorem), and a few open problems in the area. Everything presented is joint work with Luke Postle.

We discuss some recent results in graph coloring that show somewhat stronger conclusions in a similar parameter range to traditional coloring theorems. We consider the standard setup of list coloring but ask for a decomposition of the lists into pairwise-disjoint list colorings. The area is new and we discuss many open problems.

We discuss small-ball probability estimates of the smallest singular value of a rather general ensemble of random matrices which we call “inhomogeneous”. One of the novel ingredients of our family of universality results is an efficient discretization procedure, applicable under unusually mild assumption. Most of the talk will focus on explaining the ideas behind the proof of the first ingredient. Partially based on the joint work with Tikhomirov and Vershynin, and an ongoing joint work with Fernandez and Tatarko. We will also mention a related work on the cube minimal dispersion, joint with Litvak.

In this talk, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624].
 
We exploit the full symmetry of the problem by developing a block-diagonalization of the underlying matrix algebra and use it to improve bounds on several concrete instances. Our results imply that $\mathrm{cr}(K_{10,n}) \geq  4.87057 n^2 - 10n$, $\mathrm{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\mathrm{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\mathrm{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all~$n$. The latter three bounds are computed using a new relaxation of the original semidefinite programming bound, by only requiring one small matrix block to be positive semidefinite. Our lower bound on $K_{13,n}$ implies that for each fixed $m \geq 13$, $\lim_{n \to \infty} \text{cr}(K_{m,n})/Z(m,n) \geq 0.8878 m/(m-1)$. Here $Z(m,n)$ is the Zarankiewicz number: the conjectured crossing number of $K_{m,n}$.
 
This talk is based on joint work with Sven Polak.