Past

A singular perturbation analysis is presented to analyze various PDE models in a
two-dimensional domain that contain localized regions of non-uniform behavior. A
key theme of this talk is to present a unified mathematical approach, based on
an asymptotic analysis involving logarithmic series and certain Green's function
techniques, that can be used to treat a variety of PDE models such as diffusion
or eigenvalue problems in perforated domains or reaction-diffusion models with
spot-type behavior.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

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.