Fri, 18 May 2007
16:15
Martin Wood Lecture

Potassium Ion Channels

Roderick Mackinnon
(Rockefeller University (New York))
Thu, 17 May 2007
14:30
L3

TBA

Sarah Scherotzke
(Oxford)
Thu, 17 May 2007

14:00 - 15:00
Comlab

Spectral methods for PDEs in complex geometry

Prof Shiu-hong Lui
(University of Manitoba)
Abstract

Spectral methods are a class of methods for solving PDEs numerically.

If the solution is analytic, it is known that these methods converge

exponentially quickly as a function of the number of terms used.

The basic spectral method only works in regular geometry (rectangles/disks).

A huge amount of effort has gone into extending it to

domains with a complicated geometry. Domain decomposition/spectral

element methods partition the domain into subdomains on which the PDE

can be solved (after transforming each subdomain into a

regular one). We take the dual approach - embedding the domain into

a larger regular domain - known as the fictitious domain method or

domain embedding. This method is extremely simple to implement and

the time complexity is almost the same as that for solving the PDE

on the larger regular domain. We demonstrate exponential convergence

for Dirichlet, Neumann and nonlinear problems. Time permitting, we

shall discuss extension of this technique to PDEs with discontinuous

coefficients.

Thu, 17 May 2007
11:00
SR2

TBA

Alex Wilkie
(Oxford)
Tue, 15 May 2007
17:00
L1

TBA

Tue, 15 May 2007
12:00
L3

Polarized and half polarized U(1) symmetric vacuum spacetimes with AVTD behaviour.

Yvonne Choquet Bruhat
(Universite Pierre & Marie Curie)
Abstract
    I sketch, using Kichenassamy - Rendall ideas, a simplified and generalized proof of the Fuchs theorem for differential equations with a singularity. I use the theorem to construct solutions of polarized and half polarized U(1) symmetric vacuum spacetimes with "Asymptotically Velocity Terms Dominated" (AVTD) behaviour near the singularity. I show that the definition I give of half polarization for U(1) symmetric vacuum space-times is a necessary and sufficient condition for non polarized such spacetimes to have this AVTD behaviour.  
Tue, 15 May 2007
12:00
DH 1st floor SR

OCIAM /CMB Special Seminar - Holes, Traps and Spots : The Modelling and Analysis of Two-Dimensional Localised Structures in some

Professor Michael Ward
(UBC, Vancouver)
Abstract

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.

Tue, 15 May 2007
10:00
DH 2nd floor SR

Discussion Meeting 1

Abstract

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.

Mon, 14 May 2007
15:45
DH 3rd floor SR

Nonlinear Filtering of Semi-Dirichlet Processes

Professor Zhi-Ming Ma
(Chinese Academy of Sciences, Beijing)
Abstract
  The talk is based on my recent joint work with Zhechun Hu and Wei Sun. We consider a nonlinear filtering problem for general right continuous Markov processes associated with semi-Dirichlet forms. We show that in our general setting the filtering processes satisfy also DMZ (Duncan-Mortensen-Zakai) equation. The uniqueness of the solutions of the filtering equations are verified through their Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. We investigate further the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.
Mon, 14 May 2007
14:15
DH 3rd floor SR

The diameter of G (n,c/n)

Dr Oliver Riordan
(University of Cambridge (DPMS))
Abstract
  Recently, comparison with branching processes has been used to determine the asymptotic behaviour of the diameter (largest graph distance between two points that are in the same component) of various sparse random graph models, giving results for $G(n,c/n)$ as special cases. In ongoing work with Nick Wormald, we have studied $G(n,c/n)$ directly, obtaining much stronger results for this simpler model.  
Mon, 14 May 2007
12:00
L3

Self-dual supergravity and twistor theory

Martin Wolf
(Imperial College, London)
Abstract
 
By generalizing and extending some of the earlier results derived by Manin and by Merkulov, a twistor description is given of four-dimensional N-extended (gauged) self-dual supergravity with and without cosmological constant. In particular, superconformal structures are introduced and used as a starting point to define complex quaternionic, quaternionic Kaehler and hyper-Kaehler supermanifolds. A supersymmetry generalization of the Penrose and Ward constructions are presented.
 
Fri, 11 May 2007
15:15
L3

TBA

Thu, 10 May 2007

14:00 - 15:00
Comlab

Wave propagation in 1-d flexible multi-structures

Prof Enrique Zuazua
(Universidad Autonoma de Madrid)
Abstract

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.

Tue, 08 May 2007
17:00
L1

Cluster algebra structures on co-ordinate ring of flag varieties

Prof. Bernard Leclerc
(Caen)
Abstract
  Let G be a complex semisimple algebraic group of type A,D,E. Fomin and Zelevinsky conjecture that the coordinate rings of many interesting varieties attached to G have a natural cluster algebra structure. In a joint work with C. Geiss and J. Schroer we realize part of this program by introducing a cluster structure on the multi-homogeneous coordinate ring of G/P for any parabolic subgroup P of G. This was previously known only for P = B a Borel (Berenstein-Fomin-Zelevinsky) and when G/P is a grassmannian Gr(k,n) (J. Scott). We give a classification of all pairs (G,P) for which this cluster algebra has finite type. Our construction relies on a finite-dimensional algebra attached to G, the preprojective algebra introduced in 1979 by Gelfand and Ponomarev. We use the fact that the coordinate ring of the unipotent radical of P is "categorified" in a natural way by a certain subcategory of the module category of the preprojective algebra.  
Mon, 07 May 2007
17:00
L1

Energy scaling and domain branching in type-I superconductors

Sergio Conti
(Duisburg)
Abstract
  The intermediate state of a type-I superconductor is a classical example of energy-driven pattern-formation, first studied by Landau in 1937. Mathematically this can be modeled by a nonconvex functional with a singular perturbation, which physically represents the surface energy. In this talk I shall discuss how a combination of interpolation inequalities and explicit constructions permits to determine the scaling of the minimal energy with respect to the relevant material parameters, and therefore to predict a phase diagram for the observed microstructure. This talk is mainly based on joint work with Rustum Choksi, Robert V. Kohn, and Felix Otto.    
Mon, 07 May 2007
15:45
L3

Local-to-global principles for classifying spaces

Jesper Grodal
(Copenhagen)
Abstract
  In this talk I will show how one can sometimes "uncomplete" the p-completed classifying space of a finite group, to obtain the original (non-completed) classifying space, and hence the original finite group. This "uncompletion" process is closely related to well-known local-to-global questions in group theory, such as the classification of finite simple groups. The approach goes via the theory of p-local finite groups. This talk is a report on joint work with Bob Oliver.