11:00 (London time), Thursday, 09 October 2025

Imperial College London

Ryan Unger (University of California, Berkeley)

The moduli space of spherically symmetric black hole spacetimes and the extremal threshold

In this talk, we consider the moduli space of spherically symmetric solutions to the Einstein-Maxwell equations with a real scalar field. By classical work of Dafermos, it is known that solutions in this space either form black holes or disperse (no singularity forms). In the small scalar field regime, we show that the interface between these two regions of moduli space is a smooth hypersurface consisting of asymptotically extremal black holes (i.e., maximal charge for their mass). This is based on upcoming joint work with Yannis Angelopoulos (BIMSA) and Christoph Kehle (MIT).

11:00 (London time), Thursday, 02 October 2025

Imperial College London

Davide Parise (Imperial College London)

From phase transitions to minimal submanifolds in low codimensions

Proving existence of minimal submanifolds (i.e. critical points of the area functional) in various settings has been one of the driving forces of the development of modern calculus of variations, and geometric measure theory, with remarkable applications in differential geometry. In the 1960s Almgren developed a far-reaching technique to prove existence of (measure theoretic versions of) these objects. While effective, Almgren’s technique was notoriously complex which prompted the emergence in recent years of an alternative PDE-based method. The underlying idea is to construct minimal submanifolds as limits of nodal sets of critical points of functionals arising from the gradient theory of phase transitions and the theory of superconductors. After starting with a general overview, I will explain how the work of my collaborators and I fits into the broader picture. In particular, we will start with the Allen-Cahn functional and the codimension one theory discussing the construction of free boundary minimal surfaces, i.e. minimal submanifolds meeting the boundary orthogonally. We will then move to higher codimensions (specifically codimensions 2 and 3), where the theory is much less developed. We will introduce the Yang-Mills-Higgs functionals, in both the abelian and non-abelian settings. In particular, we will focus on what happens to their gradient flows, and their variational properties. I will highlight other successes of this theory, and point to some open problems along the way. The content of this talk is based on joint works with Martin Li, Lorenzo Sarnataro, Alessandro Pigati, and Daniel Stern.

12:00 (London time), Friday, 20 June 2025

Imperial College London

David Bick (University of Cambridge)

Caustic curves in the Einstein-dust system

In this talk, I will discuss the structure of singularities in the dynamics of self-gravitating pressureless fluids in General Relativity. A crucial feature is the emergence of caustic curves—envelopes formed by the trajectories of fluid particles—which arise in proposed dynamical extensions beyond these singularities.

Curvature invariants and energy densities are unbounded near such caustic curves, and so little is known rigorously about their local behaviour. I will present an existence result in a neighbourhood of a caustic curve. Specifically, a spacetime is obtained containing a caustic curve, which forms a timelike singular boundary separating a 2-dust region from a vacuum region. The spacetime arises as the solution to a PDE problem posed with a spacelike direction of evolution. Einstein's equation is satisfied weakly across the caustic. The result constitutes partial progress towards the full dynamical extension problem for shell-crossing singularities.

I will also present a novel family of static examples, complementing the construction above, which are easier to construct and analyze. Each spacetime contains an eternal 2-dust annulus bounded by a pair of caustic curves.

11:00 (London time), Friday, 20 June 2025

Imperial College London

Martin Taylor (Imperial College London)

Infalling charges and electromagnetic radiation

The Vlasov-Maxwell system describes the interaction of a large collection of collisionless charged particles. I will discuss a well posedness theorem for a "scattering problem" for this system, in which the behaviour is prescribed in the infinite past. An arbitrarily prescribed configuration of particles fall in from infinity, with the condition of no incoming electromagnetic radiation imposed for the Maxwell field. The proof is based on our previous work on the Vlasov-Poisson system. I will then discuss properties of the radiation emitted by such a configuration. This is joint work with Volker Schlue (Melbourne).

11:00 (London time), Friday, 23 May 2025

Imperial College London

Joshua Daniels-Holgate (Queen Mary University of London)

Mean curvature flow from conical singularities

We discuss some regularity results for mean curvature flow from smooth hypersurfaces with conical singularities. We then discuss how to use these results to tackle the conical singularity resolution conjecture of Ilmanen, demonstrating a non-uniqueness dichotomy: a closed flow encountering a conical singularity ‘fattens’ if and only if the asymptotic cone also fattens. This is joint work with Otis Chodosh and Felix Schulze.

11:00 (London time), Friday, 16 May 2025

Imperial College London

Peter Cameron (Imperial College London)

Spacetime extensions in low regularity

It has been shown that generic spinning black holes contain weak null singularities in their interiors. These are regions where the curvature (involving second derivatives of the metric) diverges but nonetheless spacetime can be extended as a manifold with continuous metric. Motivated by this, I will discuss uniqueness and non-uniqueness results for such continuous extensions, with the aim of showing that spacetime cannot be extended with sufficient regularity for the Einstein equations to hold. We prove this for a toy model in \(1+1\) dimensions. This represents a step towards proving the strong cosmic censorship conjecture in a neighbourhood of rotating black holes.