Past

I plan to discuss some aspects the mysterious relationship between the symmetries of toroidal compactifications of M-theory and helices on del Pezzo surfaces.

We demonstrate a scheme for controlling a large quantum system by acting

on a small subsystem only. The local control is mediated to the larger

system by some fixed coupling Hamiltonian. The scheme allows to transfer

arbitrary and unknown quantum states from a memory to the large system

("upload access") as well as the inverse ("download access").

We give sufficient conditions of the coupling Hamiltonian for the

controllability

of the system which can be checked efficiently by a colour-infection game on

the graph

that describes the couplings.

The B-D equations describe a mean field approximation for a many body system in relaxation to equilibrium. The two B-D equations determine the time evolution of the density c(L,t) of particles with mass L, L=1,2,... One of the equations is a discretized linear diffusion equation for c(L,t), and the other is a non-local constraint equivalent to mass conservation. Existence and uniqueness for the B-D system was established in the 1980's by Ball, Carr and Penrose. Research in the past decade has concentrated on understanding the large time behavior of solutions to the B-D system. This behavior is characterized by the phenomenon of "coarsening", whereby excess density is concentrated in large particles with mass increasing at a definite rate. An important conjecture in the field is that the coarsening rate can be obtained from a particular self- similar solution of the simpler LSW system. In this talk we shall discuss the B-D and LSW equations, and some recent progress by the speaker and others towards the resolution of this conjecture.

Probabilistic methods for the solution of Backward Stochastic Differential Equations (BSDE) provide us with a new approach to the problem of approximating the solution of a semi-linear PDE. Utilizing on the Markovian nature of these BSDE’s we show how one may consider the problem of numerical solutions to BSDEs within the area of weak approximations of diffusions. To emphasize this point, we suggest an algorithm based on the Cubature method on Wiener space of Lyons - Victoir. Instead of using standard discretization techniques of BSDE’s, we choose to work with the actual flow. This allows to take advantage of estimates on the derivatives of the solution of the associated semi-linear PDE and hence, we recover satisfactory convergence estimates.

I will give a few model theoretic properties for fields with a Hasse derivation which are existentially closed. I will explain how some type-definable sets allow us to understand properties of some algebraic varieties, mainly concerning their field of definition.

Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n

has a (generally branched) solution with leading order behaviour

proportional to

(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. Jointly with R.G. Halburd we consider the subclass of equations for which each possible leading order term of

this

form corresponds to a one-parameter family of solutions represented near

z_0

by a Laurent series in fractional powers of z-z_0. For this class of

equations we show that the only movable singularities that can be reached

by

analytic continuation along finite-length curves are of the algebraic type

just described. This work generalizes previous results of S. Shimomura.

The only other possible kind of movable singularity that might occur is an

accumulation point of algebraic singularities that can be reached by

analytic continuation along infinitely long paths ending at a finite point

in the complex plane. This behaviour cannot occur for constant coefficient

equations in the class considered. However, an example of R. A. Smith

shows

that such singularities do occur in solutions of a simple autonomous

second-order differential equation outside the class we consider here.

This talk shall examine a range of problems where nonlinear waves or coherent structures are localised to some portion of a domain. In one spatial dimension, the problem reduces to finding homoclinic connections to equilibria. Two canonical problems emerge when higher-order spatial terms are considered (either via fourth-order operators or discreteness effects). One involves so-called snaking bifurcation diagrams where a fundamental state grows an internal patterned layer via an infinite sequence of fold bifurcations. The other involves the exact vanishing of oscillatory tails as a parameter is varied. It is shown how both problems arise from certain codimension-two limits where they can be captured by beyond-all-orders analysis. Dynamical systems methods can then be used to explain the kind of structures that emerge away from these degenerate points. Applications include moving discrete breathers in atomic lattices, discrete solitons in optical cavities, and theories for two-dimensional localised patterns using Swift-Hohenberg theory.