I will talk about recent progress on (a) a conjecture of Fyodorov and Keating on supercritical asymptotics of moments of moments of characteristic polynomials of the circular beta ensemble and (b) on the correlation functions of the sine beta point process. This is joint work with Joseph Najnudel.

I will present a generalisation of Mirzakhani’s recursion for the volumes of moduli spaces of bordered Klein surfaces, including non-orientable surfaces. On these moduli spaces, the top form introduced by Norbury diverges as the lengths of one-sided geodesics approach zero. However, integrating this form over Gendulphe’s regularised moduli space—where the systole of one-sided geodesics is bounded below by epsilon—yields a finite volume. Using Norbury’s extension of the Mirzakhani–McShane identities to the non-orientable setting, we derive an explicit formula for the volume of the moduli space of one-bordered Klein bottles, as well as a recursion for arbitrary topologies that fully captures the dependence on the geometric regularisation parameter epsilon. I will conclude with remarks on the relation to refined topological recursion, which leads us to a refinement of the Witten–Kontsevich recursion and of the Harer–Zagier formula for the orbifold Euler characteristic of the moduli space of curves of genus g with n marked points. Based on joint work with P. Gregori and K. Osuga; the final part reflects ongoing work with N. Chidambaram, A. Giacchetto, and K. Osuga.

The cylindrical snap-fit is a ubiquitous fastening method that is both simple to manufacture and assemble, and yet secure. It consists of a partial cylindrical shell that ‘snaps’ onto a cylindrical object. We build on previous work to describe the mechanics of the cylindrical snap-fit as a naturally curved thin elastic shell placed atop a rigid cylinder; we investigate the shell's behaviour when subject to a point force pushing it onto or pulling it off the cylinder. We classify the possible contact regimes according to whether the shell has a nonzero lifting capacity. We term situations with lifting capacity ‘grip-fits’ and show that this includes both the snap-fit and a ‘stick-fit’ regime, which allows lifting despite not having the characteristic ‘snap’. We show that the different regimes may be characterized entirely by the shell/cylinder geometry and the coefficient of friction. We then consider different metrics for the lifting performance in the grip-fit regime. Our analysis reveals the trade-offs between assembly force, disassembly force, lifting force, and clamping force, providing design principles for secure lifting, easy detachment, and safe handling of fragile objects.

In the 2010s, Cornulier and Tessera presented an algorithm deciding whether a Lie group has exponential or polynomially bounded Dehn function. I will discuss the highlights of their work, and then focus on the following question: in case the Dehn function is polynomially bounded, what is the degree of the bounding polynomials? The heart of the matter in this context is the geometric relation between a (completely) solvable group and its largest nilpotent quotient. I will outline the basics of this geometry, and present a new method that exploits it to give (in some cases) better bounds on the degree of the bounding polynomials.

Joint with Gabriel Pallier.

Clean markings on surfaces were a key component in Masur and Minsky's hierarchy machinery, which proved to be a powerful tool in the study of mapping class groups. In this talk, I will briefly discuss the connection between clean markings and hierarchies, and I will explain how a natural analogue can be constructed for finite-type Artin groups.

A Groethendieck pair consists of a finitely generated residually finite group G, with a finitely generated subgroup N such that the inclusion N -> G induces an isomorphism of profinite completions. I will present a new method to produce Groethendieck pairs with peculiar properties, using iterated group theoretic Dehn filling on hyperbolic virtually special groups. Such pairs witness the profinite non-invariance of quasimorphisms, stable commutator length, and actions on hyperbolic spaces and finite-dimensional CAT(0) cube complexes.

Let G be a group acting on a tree. I will discuss necessary conditions for G to have a finitely generated infinite normal subgroup of infinite index. When the edge stabilisers are virtually cyclic this naturally leads to considering (virtual) fibering of G. I will give an “if and only if” criterion for (virtual) fibering in the special case of amalgamated free products over virtually cyclic subgroups. The talk will be based on joint work with Jon Merladet.