28 September 2015

Speaker: Kirill Cherednichenko (Bath).

Title: Boundary triples, Krein formula, and resolvent estimates for one-dimensional high-contrast periodic problems

Abstract: I will discuss operator-norm resolvent convergence estimates for one-dimensional periodic differential operators with rapidly oscillating coefficients in the non-uniformly elliptic high-contrast setting, which has been out of reach of the existing homogenisation techniques. Our analysis is based on a special representation of the resolvent of the operator in terms of the $M$-matrix of an associated boundary triple, due to M. G. Krein. The resulting asymptotic behaviour is shown to be described, up to a unitary equivalent transformation, by a non-standard version of the Kronig-Penney model on $\mathbb R$. This is joint work with Alexander V. Kiselev.

5 October 2015

Speaker: Alexander V. Sobolev (University College London).

Title: On non-smooth functions of Wiener-Hopf operators

Abstract: We discuss trace formulae for the operator $f(PAP) - Pf(A)P$, where $A$ is a pseudo-differential operator on $L^2(R^d)$ with a smooth or discontinuous symbol, and $P$ is a multiplication by the indicator of a piece-wise smooth domain in $R^d$. The function $f$ is not supposed to be smooth. The obtained formulae generalise results obtained by H. Widom in the 80's.These results are used to study the entanglement entropy of free fermions at positive temperature both in the low and high temperature limits.

12 October 2015

Speaker: Colin Guillarmou (ENS-Paris).

Title: TBC

Abstract: TBC

19 October 2015

Speaker: Yuri Korolev (Queens Mary London).

Title: Regularisation of Linear Ill-posed Problems in Banach Lattices

Abstract: Partial order is a handy tool in deterministic error modelling. Many functional spaces that are common in the practice of solving inverse problems, such as the L^p-spaces, can be equipped with a natural partial order relation, which turns them into Banach lattices - functional spaces with a partial order and a norm that agree with each other in a certain sense.

Partial order can be used in the formulation of an inverse problem to represent errors in the data and the forward operator as bounds by means of the appropriate partial orders. One advantage is that one gets convex fidelity constraints even if the operator is not known exactly - something one does not get with a norm only.

In this talk, I will present some theoretical results about regularisation in this novel framework and discuss some computational aspects. Time permitting, I will also briefly discuss an application in diffusion tensor imaging.

26 October 2015

Speaker: Suresh Eswarathasan (Cardiff).

Title: TBC

Abstract: TBC

2 November 2015

Speaker: Goeffrey Burton (Bath).

Title: Variational problems involving rearrangements of functions.

Abstract: Two real functions on a domain in Euclidean space are said to be rearrangements of one another if they have the same decreasing rearrangement, that is, corresponding super-level sets of the two functions always have equal measure.
Variational problems where the set of all rearrangements of one fixed function forms the constraint arise in fluid mechanics, where the maximisers of kinetic energy represent steady configurations for a vortex.
Existence of maximisers can be proved by means of convex analysis in conjunction with compactness properties of highly symmetric functions. In order to study stability one needs to prove compactness of all maximising sequences, which may lack symmetry. This necessitates the use of concentration-compactness.

9 November 2015

Speaker: Katie Gittins (Bristol).

Title: TBC

Abstract: TBC

16 November 2015

Speaker: Valery Smyshlyaev (University College London).

Title: TBC

Abstract: TBC

23 November 2015

Speaker: Alessio Martini (Birmingham).

Title: TBC

Abstract: TBC

1 February 2015

Speaker: Jiang-Lun Wu (Swansea).

Title: TBC

Abstract: TBC

7 March 2015

Speaker: Jonathan Ben-Artzi (Imperial College London).

Title: TBC

Abstract: TBC