University College London

I will explain some recent joint work with Georgios Dimitroglou Rizell in which we use moduli spaces of holomorphic discs with boundary on a monotone Lagrangian torus in ${\mathbb C}^n$ to prove that all such tori are smoothly isotopic when $n$ is odd and at least 5

We discuss the problem of estimating a structured matrix with a large number of elements. A key motivation for this problem occurs in multi-task learning. In this case, the columns of the matrix correspond to the parameters of different regression or classification tasks, and there is structure due to relations between the tasks. We present a general method to learn the tasks' parameters as well as their structure. Our approach is based on solving a convex optimization problem, involving a data term and a penalty term. We highlight different types of penalty terms which are of practical and theoretical importance. They implement structural relations between the tasks and achieve a sparse representations of parameters. We address computational issues as well as the predictive performance of the method. Finally we discuss how these ideas can be extended to learn non-linear task functions by means of reproducing kernels.

Nonnormality is a well studied subject in the context of partial differential operators. Yet, only little is known for boundary integral operators. The only well studied case is the unit ball, where the standard single layer, double layer and conjugate double layer potential operators in acoustic scattering diagonalise in a unitary basis. In this talk we present recent results for the analysis of spectral decompositions and nonnormality of boundary integral operators on more general domains. One particular application is the analysis of stability constants for boundary element discretisations. We demonstrate how these are effected by nonnormality and give several numerical examples, illustrating these issues on various domains.