Past

  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.

 

 

Diffusion limited aggregation (DLA) is a random growth model which was

originally introduced in 1981 by Witten and Sander. This model is prevalent in

nature and has many applications in the physical sciences as well as industrial

processes. Unfortunately it is notoriously difficult to understand, and only one

rigorous result has been proved in the last 25 years. We consider a simplified

version of DLA known as the Eden model which can be used to describe the growth

of cancer cells, and show that under certain scaling conditions this model gives

rise to a limit object known as the Brownian web.

 

We study the parabolic Anderson problem, i.e., the heat equation on the d-dimentional

integer lattice with independent identically distributed random potential and

localised initial condition. Our interest is in the long-term behaviour of the

random total mass of the unique non-negative solution, and we prove the complete

localisation of mass for potentials with polynomial tails.