Thu, 07 Nov 2019
16:00
L6

Number fields with prescribed norms

Rachel Newton
(Reading)
Abstract

Let G be a finite abelian group, let k be a number field, and let x be an element of k. We count Galois extensions K/k with Galois group G such that x is a norm from K/k. In particular, we show that such extensions always exist. This is joint work with Christopher Frei and Daniel Loughran.

Tue, 22 May 2018
17:00
C1

On the spectral resolution of the Neumann-Poincare operator

Karl-Mikael Perfekt
(Reading)
Abstract

The Neumann-Poincare (NP) operator (or the double layer potential) has classically been used as a tool to solve the Dirichlet and Neumann problems of a domain. It also serves as a prominent example in non-self adjoint spectral theory, due to its unexpected behaviour for domains with singularities. Recently, questions from materials science have revived interest in the spectral properties of the NP operator on domains with rough features. I aim to give an overview of recent developments, with particular focus on the NP operator's action on the energy space of the domain. The energy space framework ties together Poincare’s efforts to solve the Dirichlet problem with the operator-theoretic symmetrisation theory of Krein. I will also indicate recent work for domains in 3D with conical points. In this situation, we have been able to describe the spectrum both for boundary data in $L^2$ and for data in the energy space. In the former case, the essential spectrum consists of the union of countably many self-intersecting curves in the plane, and outside of this set the index may be computed as the winding number with respect to the essential spectrum. In the latter case the essential spectrum consists of a real interval.

Tue, 07 Feb 2017
17:00
C1

Banach-Stone type theorems on spaces of probability measures

Gyorgy Geher
(Reading)
Abstract

The classical Banach-Stone theorem describes the structure of onto linear isometries of the Banach space $C(K)$ of all continuous functions on a compact Hausdorff space $K$. Namely, such an isometry is always a product of a composition operator with a homeomorphism symbol and a multiplication operator with a continuous symbol which has modulus 1.

Recently, similar results have been obtained in the setting of certain class of probability measures. In my talk first, I will give an overview of these results, and then I will present the main ideas of a recent work. Namely, I will provide a characterisation of all surjective isometries of the (non-linear) space of all Borel probability measures on an arbitrary separable Banach space with respect to the famous Levy-Prokhorov distance (which metrises the weak convergence). This is a recent joint work with Tamas Titkos (MTA Alfred Renyi Institute of Mathematics, Budapest, Hungary).

Tue, 12 Nov 2013

17:00 - 18:12
C6

The heat equation in curved stripes

Martin Kolb
(Reading)
Abstract

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the flat case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the flat case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.

Thu, 06 Mar 2014

16:00 - 17:00
L3

The effect of boundary conditions on linear and nonlinear waves

Beatrice Pelloni
(Reading)
Abstract

In this talk, I will discuss the effect of boundary conditions on the solvability of PDEs that have formally an integrable structure, in the

sense of possessing a Lax pair. Many of these PDEs arise in wave propagation phenomena, and boundary value problems for these models are very important in applications. I will discuss the extent to which general approaches that are successful for solving the initial value problem extend to the solution of boundary value problem.

I will survey the solution of specific examples of integrable PDE, linear and nonlinear. The linear theory is joint work with David Smith. For the nonlinear case, I will discuss boundary conditions that yield boundary value problems that are fully integrable, in particular recent joint results with Thanasis Fokas and Jonatan Lenells on the solution of boundary value problems for the elliptic sine-Gordon equation.

Tue, 17 Jan 2012

17:00 - 18:47
L3

Random Tri-Diagonal Operators

Simon Chandler-Wilde
(Reading)
Abstract

In this talk I will describe recent work by myself and others (E.B. Davies (KCL), M. Lindner (Chemnitz), S. Roch (Darmstadt)) on the spectrum and essential spectrum of bi-infinite and semi-infinite (not necessarily self-adjoin) tri-diagonal random operators, and the implications of these results for the spectra of associated random matrices, and for the finite section method for infinite tri-diagonal systems. A main tool will be limit operator methods, as described in Chandler-Wilde and Lindner, Memoirs AMS, 2011), supplemented by certain symmetry arguments including a Coburn lemma for random matrices.

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