Past

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.

 

For a given Lipschitz graph over a subspace without rank-one matrices with

reasonably small Lipschitz constant, we construct quasiconvex functions of

quadratic growth whose zero sets are exactly the Lipschitz graph by using a

translation method. The gradient of the quasiconvex function is strictly

quasi-monotone. When the graph is a smooth compact manifold, the quasiconvex

function equals the squared distance function near the graph.

The corresponding variational integrals satisfy the Palais-Smale compactness

condition under the homogeneous natural boundary condition.