Past

Tonelli gave the first rigorous treatment of one-dimensional variational problems, providing conditions for existence and regularity of minimizers over the space of absolutely continuous functions.  He also proved a partial regularity theorem, asserting that a minimizer is everywhere differentiable, possible with infinite derivative, and that this derivative is continuous as a map into the extended real line.  Some recent work has lowered the smoothness assumptions on the Lagrangian for this result to various Lispschitz and H\"older conditions.  In this talk we will discuss the partial regularity result, construct examples showing that mere continuity of the Lagrangian is an insufficient condition.

Study the parabolic Hitchin fibrations for Langlands dual groups. Sketch the proof of a duality theorem of the natural symmetries on their cohomology.

There will be three problems discussed all of which are open for consideration as MSc projects.

1. Reduction of Ndof in Adaptive Signal Processing

2. Calculus of Convex Sets

3. Dynamic Response of a disc with an off centre hole(s)

In my talk I will introduce the concept of spectral discrete solitons

(SDSs): solutions of nonlinear Schroedinger type equations, which are localized on a regular grid in frequency space. In time domain such solitons correspond to periodic trains of pulses. SDSs play important role in cascaded four-wave-mixing processes (frequency comb generation) in optical fibres, where initial excitation by a two-frequency pump leads to the generation of multiple side-bands. When free space diffraction is taken into consideration, a non-trivial generalization of 1D SDSs will be discussed, in which every individual harmonic is an optical vortex with its own topological charge. Such excitations correspond to spatio-temporal helical beams.

We propose and analyze a primal-dual active set method for local and non-local vector-valued Allen-Cahn variational inequalities.

We show existence and uniqueness of a solution for the non-local vector-valued Allen-Cahn variational inequality in a formulation involving Lagrange multipliers for local and non-local constraints. Furthermore, convergence of the algorithm is shown by interpreting the approach as a semi-smooth Newton method and numerical simulations are presented.

All of Joyce's constructions of compact manifolds with special holonomy are in some sense generalisations of the Kummer construction of a K3 surface. We will begin by reviewing manifolds with special holonomy and the Kummer construction. We will then describe Joyce's constructions of compact manifolds with holonomy G_2 and Spin(7).

Extend the affine Weyl group action in Lecture I to double affine Hecke algebra action, and (hopefully) more examples.

Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.