In this talk, I will describe a family of observables for 3D quantum Yang-Mills theory based on regularising connections with the YM heat flow. I will describe how these observables can be used to show that there is a unique renormalisation of the stochastic quantisation equation of YM in 3D that preserves gauge symmetries. This complements a recent result on the existence of such a renormalisation. Based on joint work with Hao Shen.

Do there exist universal sequences for all mazes on the two-dimensional integer lattice? We will give background on this question, as well as some recent results. Joint work with Mariaclara Ragosta.

Semidefinite programming (SDP) is a powerful tool in the design of approximation algorithms. After providing a gentle introduction to the basics of this method, I will explore a different facet of SDP and show how it can be used to derive short and elegant proofs of both classical and new estimates related to the MaxCut problem and discrepancy theory in graphs and matrices.

Building on this, I will demonstrate how these results lead to an improved upper bound on the celebrated log-rank conjecture in communication complexity.

A matroid is frame if it can be extended such that it possesses a basis $B$ (a frame) such that every element is spanned by at most two elements of $B$. Frame matroids extend the class of graphic matroids and also have natural graphical representations. We characterise the inequivalent graphical representations of 3-connected frame matroids that have a fixed element $\ell$ in their frame $B$. One consequence is a polynomial time recognition algorithm for frame matroids with a distinguished frame element.

Joint work with Jim Geelen and Cynthia Rodríquez.

At most how many edge-disjoint Hamilton cycles does a given directed graph contain? It is easy to see that one cannot pack more than the minimum in-degree or the minimum out-degree of the digraph. We show that in the random directed graph $D(n,p)$ one can pack precisely this many edge-disjoint Hamilton cycles, with high probability, given that $p$ is at least the Hamiltonicity threshold, up to a polylog factor.

Based on a joint work with Asaf Ferber.

Graphs (and structures) which have a linear ordering of their vertices with given local properties have a rich spectrum of complexities. Some have full power of class NP (and thus no dichotomy) but for biconnected patterns we get dichotomy. This also displays the importance of Sparse Incomparability Lemma. This is a joint work with Gabor Kun (Budapest).

Cryptocurrencies such as Bitcoin have only recently become a significant part of the financial landscape. Many billions of dollars are now traded daily on limit order-book markets such as Binance, and these are probably among the most open, liquid and transparent markets there are. They therefore make an interesting platform from which to investigate myriad questions to do with market microstructure. I shall talk about a few of these, including live-trading experiments to investigate the difference between on-paper strategy analysis (typical in the academic literature) and actual trading outcomes. I shall also mention very recent work on the new Hyperliquid exchange which runs on a blockchain basis, showing how to use this architecture to obtain datasets of an unprecendented level of granularity. This is joint work with Jakob Albers, Mihai Cucuringu and Alex Shestopaloff.