The seminars take place on Wednesdays from 4.30 - 5.30 pm in Roger Stevens LT 14
Black holes and scattering resonances
Recent experiments have, for the first time, directly measured gravitational waves created by colliding black holes. An important part of the signal from such events is the 'ringdown’ phase where a distorted black hole emits radiation at certain fixed (complex) frequencies called the quasinormal frequencies. To mathematically model this phenomenon, one should study geometric wave equations on a class of open geometries. I will discuss how the quasinormal frequencies can be realised as eigenvalues of a (non-standard) spectral problem, with connections to scattering resonances on asymptotically hyperbolic manifolds. If time permits I will also discuss recent work with Gajic on the asymptotically flat case.
Finite-time degeneration for Teichmüller harmonic map flow
Teichmüller harmonic map flow is a geometric flow designed to evolve combinations of maps and metrics on a surface into minimal surfaces in a Riemannian manifold. I will introduce the flow and describe known existence results, and discuss recent joint work with M. Rupflin that demonstrates how singularities can develop in the metric component in finite time.
On Laplace--Carleson embeddings
The talk is based on a work in progress, and deals with $L^p$ mapping properties of the one-sided Laplace transform. After a brief recollection of previous work by Jacob, Partington and Pott, I present a new result on so-called Laplace--Carleson embeddings. The main novelty of my result is that it extends beyond the regime of the Hausdorff--Young theorem. I will also discuss a simple connection to a result by Hardy and Littlewood. Given its simplicity, the natural formulation for this talk seems surprisingly difficult to find in the literature.
Attractors of the Einstein-Klein-Gordon system
A key question in general relativity is whether solutions to the Einstein equations, viewed as an initial value problem, are stable to small perturbations of the initial data. For example, previous results have shown that the Milne spacetime, which represents an expanding universe emanating from a big bang singularity with a linear scale factor, is a stable solution to the Einstein equations. With such a slow expansion rate, particularly compared to related isotropically expanding models (such as the exponentially expanding de Sitter spacetime observed in our universe), there are interesting questions one can ask about stability of this spacetime. Previous results have shown that the Milne model is a stable solution to the vacuum Einstein, Einstein-Klein-Gordon and Einstein-Vlasov systems. Motivated by techniques from the last result, I will present a new proof of the stability of the Milne model to the Einstein-Klein-Gordon system and compare our method to a recent result of J. Wang. This is joint work with D. Fajman.
Functional calculus for analytic Besov functions
I shall describe work with Alexander Gomilko and Yuri Tomilov in which we develop a bounded functional calculus for analytic Besov functions applicable to unbounded operators, sepcifically generators of many bounded semigroups, including bounded semigroups on Hilbert spaces and bounded holomorphic semigroups on Banach spaces. The calculus is a natural extension of the classical Hille-Phillips functional calculus, and it is compatible with the other well-known functional calculi. It satisfies the standard properties of functional calculi, provides a unified and direct approach to a number of norm-estimates in the literature, and enables improvements of some of them.