Past

 

Relatively little is known about the ability of numerical methods for stochastic differential equations (SDE

 

Abstract:

(joint work with Allen Hatcher) Let M be a compact, connected 3-manifold with a

fixed boundary component d_0M. For each prime manifold P, we consider the

mapping class group of the manifold M_n^P obtained from M by taking a connected

sum with n copies of P. We prove that the ith homology of this mapping class

group is independent of n in the range n>2i+1. Our theorem moreover applies to

certain subgroups of the mapping class group and include, as special cases,

homological stability for the automorphism groups of free groups and of other

free products, for the symmetric groups and for wreath products with symmetric

groups.

 

I shall report on a programme of research which is joint with Terence Tao. Our

goal is to count the number of solutions to a system of linear equations, in

which all variables are prime, in as much generality as possible. One success of

the programme so far has been an asymptotic for the number of four-term

arithmetic progressions p_1 < p_2 < p_3 < p_4 <= N of primes, defined by the

pair of linear equations p_1 + p_3 = 2p_2, p_2 + p_4 = 2p_3. The talk will be

accessible to a general audience.

  I shall describe some joint work with Vladimir Remeslennikov and Ilia Kazachkov. Partially commutative groups are groups given by a presentation determined by a graph: vertices are generators and edges define commutation relations. Divisbility and orthogonal systems are tools developed to study these groups. Using them we have descriptions of centralisers of subsets, a good understanding of the centraliser lattice in terms of the underlying graph and have made good progress towards classifying the universal theory of these groups as well as their automorphism groups.

The "secular equation" is a special way of expressing eigenvalue

problems in a variety of applications. We describe the secular

equation for several problems, viz eigenvector problems with a linear

constraint on the eigenvector and the solution of eigenvalue problems

where the given matrix has been modified by a rank one matrix. Next we

show how the secular equation can be approximated by use of the

Lanczos algorithm. Finally, we discuss numerical methods for solving

the approximate secular equation.