A branched covering between two surfaces looks like a regular covering map except for finitely many branching points, where some non-trivial ramification may occur. Informally speaking, the existence problem asks whether we can find a branched covering with prescribed behaviour around its branching points.

A variety of techniques have historically been employed to tackle this problem, ranging from studying representations of surface groups into symmetric groups to drawing "dessins d'enfant" on the covering surface. After introducing these techniques and explaining how they can be applied to the existence problem, I will briefly present a conjecture unexpectedly relating branched coverings and prime numbers.
 

This will be a gentle introduction to the theory of pseudo-Anosov  flows on 3-manifolds, as seen from the perspective of a topologist and not a dynamicist.

I will start by considering geodesic flows on the unit tangent bundles of hyperbolic surfaces. This will lead to a definition of an Anosov and then a pseudo-Anosov flow on a 3-manifold. After discussing a couple of examples, I will outline some connections between pseudo-Anosov flows and other aspects of 3-manifold topology/ geometry/ group theory.

We will give a fairly self contained introduction to acylindrically hyperbolic groups, using mapping class groups as a motivating example. This will be a mainly expository talk, the aim is to make my topology seminar talk in week 5 more accessible to people who are less familiar with these topics.

G2 structure manifolds are a key ingredient in supersymmetric compactifications on seven-manifolds. We will discuss the fact that G2 structure manifolds admit refinements in the form of almost contact (3-) structures.  In fact, there are infinite dimensional spaces of these structures. We will discuss topological and differential geometric aspects of (the space of) these refinements. We will then explore applications in physics, including supersymmetry enhancement. This is based on 2101.12605.

I will report on my most recent results  with Jeremie Szeftel and Elena Giorgi which conclude the proof of the nonlinear, unconditional, stability of slowly rotating Kerr metrics. The main part of the proof, announced last year, was conditional on results concerning boundedness and decay estimates for nonlinear wave equations. I will review the old results and discuss how the conditional results can now be fully established.