Past

I will discuss the so-called Lasso method for signal recovery for high-dimensional data and show applications in computational biology, machine learning and image analysis.

This talk will be based on a joint work with Professor Terry Lyons and Mr Gechun Liang (OMI). I will explain a new approach to define and to solve a class of backward dynamic systems including the well known examples of non-linear backward SDE. The new approach does not require any kind of martingale representation or any specific restriction on the probability base in question, and therefore can be applied to a much wider class of backward systems.

I will briefly describe a differential geometric construction of Hitchin's projectively flat connection in the Verlinde bundle, over Teichm\"uller space, formed by the Hilbert spaces arising from geometric quantization of the moduli space of flat connections on a Riemann surface. We will work on a general symplectic manifold sharing certain properties with the moduli space. Toeplitz operators enter the picture when quantizing classical observables, but they are also closely connected with the notion of deformation quantization. Furthermore, through an intimate relationship between Toeplitz operators, the Hitchin connection manifests itself in the world of deformation quantization as a connection on formal functions. As we shall see, this formal Hitchin connection can be used to construct a deformation quantization, which is independent of the Kähler polarization used for quantization. In the presence of a symmetry group, this deformation quantization can (under certain cohomological conditions) be constructed invariantly. The talk presents joint work with J. E. Andersen.

For incompressible Navier-Stokes equations in a bounded domain, I will

first present a formula for the pressure that involves the commutator

of the Laplacian and Leray-Helmholtz projection operators. This

commutator and hence the pressure is strictly dominated by the viscous

term at leading order. This leads to a well-posed and computationally

congenial unconstrained formulation for the Navier-Stokes equations.

Based on this pressure formulation, we will present a new

understanding and design principle for third-order stable projection

methods. Finally, we will discuss the delicate inf-sup stability issue

for these classes of methods. This is joint work with Bob Pego and Jie Liu.

A new class of relativistic diffusions encompassing all the previously studied examples has recently been introduced by C. Chevalier and F Debbasch, both in a heuristic and analytic way.  Roughly speaking, they are characterised by the existence at each (proper) time (of the moving particle) of a (local) rest frame where the random part of the acceleration of the particle (computed using the time of the rest frame) is brownian in any spacelike direction of the frame.

I will explain how the tools of stochastic calculus enable us to give a concise and elegant description of these random paths on any Lorentzian manifiold.  A mathematically clear definition of the the one-particle distribution function of the dynamics will emerge from this definition, and whose main property will be explained.  This will enable me to obtain a general H-theorem and to shed some light on links between probablistic notions and the large scale structure of the manifold.

All necessary tools from stochastic calculus and geometry will be explained.

Abstract:  Modern techniques for computing multi-particle and multi-loop scattering amplitudes rely on a sophisticated use of on-shell recursion relations and generalised unitarity methods. I will show that these methods are ideally suited to interpretation in twistor space, where superconformal properties become manifest. In fact, the recursion relations of Britto, Cachazo, Feng & Witten provide a clear framework for the twistor diagram program initiated in the 1970s.
Tree-level scattering amplitudes in N=4 SYM are now known to possess a Yangian symmetry, formed by combining the original PSU(2,2|4) superconformal invariance with a second "dual" copy. I will also discuss very recent work constructing scattering amplitudes in a twistor space in which this dual superconformal symmetry acts geometrically.

Soliton like structures called “stable droplets” are found to exist within a paradigm reaction
diffusion model which can be used to describe the patterning in a number of fish species. It is
straightforward to analyse this phenomenon in the case when two non-zero stable steady states are
symmetric, however the asymmetric case is more challenging. We use a recently developed
perturbation technique to investigate the weakly asymmetric case.