Past

I will speak about a geometric method, based on the classical trace map, for investigating word maps on groups PSL(2, q) and SL(2, q). In two different papers (with F. Grunewald, B. Kunyavskii, and Sh. Garion, F. Grunewald, respectively) this approach was applied to the following problems.

1. Description of Engel-like sequences of words in two variables which characterize finite

solvable groups. The original problem was reformulated in the language of verbal dynamical

systems on SL(2). This allowed us to explain the mechanism of the proofs for known

sequences and to obtain a method for producing more sequences of the same nature.

2. Investigation of the surjectivity of the word map defined by the n-th Engel word

[[[X, Y ], Y ], . . . , Y ] on the groups PSL(2, q) and SL(2, q). Proven was that for SL(2, q), this

map is surjective onto the subset SL(2, q) $\setminus$ {−id} $\subset$ SL(2, q) provided that q $\ge q_0(n)$ is

sufficiently large. If $n\le 4$ then the map was proven to be surjective for all PSL(2, q).

Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms.

In a healthy human brain, cerebrospinal fluid (CSF), a water-like liquid, fills a system of cavities, known as ventricles, inside the brain and also surrounds the brain and spinal cord. Abnormalities in CSF dynamics, such as hydrocephalus, are not uncommon and can be fatal for the patient. We will consider two types of models for the so-called infusion test, during which additional fluid is injected into the CSF space at a constant rate, while measuring the pressure continuously, to get an insight into the CSF dynamics of that patient.

In compartment type models, all fluids are lumped into compartments, whose pressure and volume interactions can be modelled with compliances and resistances, equivalent to electric circuits. Since these models have no spatial variation, thus cannot give information such as stresses in the brain tissue, we also consider a model based on the theory of poroelasticity, but including strain-dependent permeability and arterial blood as a second fluid interacting with the CSF only through the porous elastic solid.

We show the existence of global-in-time weak solutions to a general class of bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.

The talk is based on joint work with John W. Barrett [Imperial College London].

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).