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.
Floating mats and sloping beaches: spectral asymptotics of the Steklov problem on polygons
I will discuss the asymptotic behaviour of the eigenvalues of the Steklov problem (aka Dirichlet-to-Neumann operator) on curvilinear polygons. The answer is completely unexpected and depends on the arithmetic properties of the angles of the polygon.
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.
Abstract structure of Banach function algebras on homogeneous spaces of compact groups
I will discuss abstract structure of classical Banach function *-algebras on homogeneous spaces (coset spaces) of compact groups. Suppose $G$ is a compact group and $H$ is a closed subgroup of $G$. Let $G/H$ be the left coset space of $H$ in $G$ and $\mu$ be the $G$-invariant measure on the homogeneous space $G/H$ normalized with respect to Weil's formula. I shall talk about the abstract notions of convolution and involution for functions in $L^p(G/H,\mu)$, for all $p\ge 1$.
The essential numerical range of the Laplace double-layer potential on Lipschitz domains
This talk is concerned with the theory of boundary integral equations for Laplace's equation on Lipschitz domains.
The theory for these equations in the space $L^2(\Gamma)$, where $\Gamma$ is the boundary of the domain, was developed in the 1980s by, e.g., Calderon, Coifman, McIntosh, Meyer, and Verchota.
However, the following question has remained open: can the standard second-kind integral equations, posed in $L^2(\Gamma)$, be written as the sum of a coercive operator plus a compact operator when $\Gamma$ is only assumed to be Lipschitz, or even Lipschitz polyhedral?
These second-kind equations involve the double-layer potential and this “coercive + compact” property can be rephrased as the property that the essential numerical range of the double-layer potential does not contain plus or minus one.
The practical importance of this question is that the convergence analysis of Galerkin discretisations of these integral equations relies on this “coercive + compact” property holding.
This talk will describe joint work with Simon Chandler-Wilde (University of Reading) that answers this question.
The spectral density of self-adjoint Hankel Matrices
In 1966, H. Widom studied the distribution of eigenvalues of the NxN Hilbert matrix and showed that it diverges logarithmically in N. In this talk, we will generalise his result to any self-adjoint Hankel matrix with a piece-wise continuous symbol.