Past

  I am going to discuss a special class of logarithmic solutions to WDVV equations. This type of solutions appeared in Seiberg-Witten theory is defined by a finite set of covectors, the V-systems. The V-systems introduced by Veselov have remarkable properties. They contain Coxeter root systems, and they are closed under taking subsystems and restrictions. The corresponding solutions are almost dual in Dubrovin's sense to the Frobenius manifolds structures on the orbit spaces of Coxeter groups and their restrictions to discriminants. Another source of V-systems is generalized root systems. The talk will be based on joint work with Veselov.    

  A dimer configuration of a graph is a subset of the edges, such that every vertex is contained in exactly one edge of the subset. We consider dimer configurations of the honeycomb lattice on the cylinder, which are known to be equivalent to configurations of interlaced particles. Assigning a measure to the set of all such configurations, we show that the probability that particles are located in any subset of points on the cylinder can be written as a determinant, i.e. that the process is determinantal. We also examine Markov chains of interlaced particles on the circle, with dynamics equivalent to RSK.  

 

Gradient bounds are a very powerful tool to study heat kernel measures and

regularisation properties for the heat kernel. In the elliptic case, it is easy

to derive them from bounds on the Ricci tensor of the generator. In recent

years, many efforts have been made to extend these bounds to some simple

examples in the hypoelliptic situation. The simplest case is the Heisenberg

group. In this talk, we shall discuss some recent developments (due to H.Q. Li)

on this question, and give some elementary proofs of these bounds.

 

  The "Cubature on Wiener space" algorithm can be regarded as a general approach to high order weak approximations. Based on this observation we will derive many well known weak discretisation schemes and optimise the computational effort required for a given accuracy of the approximation. We show that cubature can also help to overcome some stability difficulties. The cubature on Wiener space algorithm is frequently combined with partial sampling techniques and we outline an extension to these methods to reduce the variance of the samples. We apply the extended method to examples arising in mathematical finance. Joint work of G. Gyurko, C. Litterer and T. Lyons  

Spectral methods are a class of methods for solving PDEs numerically.

If the solution is analytic, it is known that these methods converge

exponentially quickly as a function of the number of terms used.

The basic spectral method only works in regular geometry (rectangles/disks).

A huge amount of effort has gone into extending it to

domains with a complicated geometry. Domain decomposition/spectral

element methods partition the domain into subdomains on which the PDE

can be solved (after transforming each subdomain into a

regular one). We take the dual approach - embedding the domain into

a larger regular domain - known as the fictitious domain method or

domain embedding. This method is extremely simple to implement and

the time complexity is almost the same as that for solving the PDE

on the larger regular domain. We demonstrate exponential convergence

for Dirichlet, Neumann and nonlinear problems. Time permitting, we

shall discuss extension of this technique to PDEs with discontinuous

coefficients.