We give an exponential improvement on the upper bound for the $r$-colour diagonal Ramsey number for all $r$. The proof relies on geometric insights and offers a simplified proof in the case of $r=2$.

Joint Work with: Paul Ballister, Béla Bollobás, Marcelo Campos, Simon Griffiths, Rob Morris, Julian Sahasrabudhe and Marius Tiba.

We show that an $n$-vertex $k$-uniform hypergraph, where all $(k-1)$-subsets that are supported by an edge are in fact supported by at least $n/2+o(n)$ edges, contains a spanning $(k-1)$-dimensional sphere. This generalises Dirac's theorem, and confirms a conjecture of Georgakopoulos, Haslegrave, Montgomery, and Narayanan. Unlike typical results in the area, our proof does not rely on the absorption method or the regularity lemma. Instead, we use a recently introduced framework that is based on covering the vertex set of the host hypergraph with a family of complete blow-ups.

This is joint work with Freddie Illingworth, Richard Lang, Olaf Parczyk, and Amedeo Sgueglia.

One recent(ish) development in classical black hole thermodynamics is the inclusion of vacuum energy (cosmological constant) in the form of thermodynamic pressure. New thermodynamic phase transitions emerge in this extended phase space, beyond the usual Hawking—Page transition. This allows us to understand black holes from the viewpoint of chemistry in terms of concepts such as Van Der Waals fluids, reentrant phase transitions and triple points. I will review these developments and discuss the dictionary between the bulk laws and those of the dual CFT.
 

I will explain the recent techniques developed with co-authors to obtain fine estimates about the large values of the Riemann zeta functions on the critical line. An emphasis will be put on the ideas originating from statistical mechanics and large deviations that may be of general interest for a stochastic analysis audience. No number theory knowledge will be assumed!

We propose a natural version of Connes' Rigidity Conjecture (1982) that involves property (T) groups with infinite centre. Using methods at the rich intersection between von Neumann algebras and geometric group theory, we identify several instances where this conjecture holds. This is joint work with Ionut Chifan, Denis Osin, and Hui Tan.