Past

In this talk we will mainly analyze the vibrations of a simplified 1-d model for a multi-body structure consisting of a finite number of flexible strings distributed along a planar graph. In particular we shall analyze how solutions propagate along the graph as time evolves. The problem of the observation of waves is a natural framework to analyze this issue. Roughly, the question can be formulated as follows: Can we obtain complete information on the vibrations by making measurements in one single extreme of the network? This formulation is relevant both in the context of control and inverse problems.

Using the Fourier development of solutions and techniques of Nonharmonic Fourier Analysis, we give spectral conditions that guarantee the observability property to hold in any time larger than twice the total lengths of the network in a suitable Hilbert that can be characterized in terms of Fourier series by means of properly chosen weights. When the network graph is a tree these weights can be identified.

Once this is done these results can be transferred to other models as the Schroedinger, heat or beam-type equations.

This lecture is based on results obtained in collaboration with Rene Dager.

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.

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.