Tue, 24 Feb 2015

14:00 - 14:30
L5

A hybrid numerical-asymptotic boundary element method for scattering by penetrable obstacles

Samuel Groth
(University of Reading)
Abstract

When high-frequency acoustic or electromagnetic waves are incident upon an obstacle, the resulting scattered field is composed of rapidly oscillating waves. Conventional numerical methods for such problems use piecewise-polynomial approximation spaces which are not well-suited to capture the oscillatory solution. Hence these methods are prohibitively expensive in the high-frequency regime. Much work has been done in developing “hybrid numerical-asymptotic” (HNA) boundary element methods which utilise approximation spaces containing oscillatory functions carefully chosen to capture the high-frequency asymptotic behaviour of the solution. The computational cost of this approach is significantly smaller than that of conventional methods, and for many problems it is independent of the frequency. In this talk, I will outline the HNA method and discuss its extension to scattering by penetrable obstacles.​

Thu, 12 Feb 2015
11:00
C5

Matrix multiplication is determined by orthogonality and trace.

Chris Heunen
(Oxford)
Abstract

Everything measurable about a quantum system, as modelled by a noncommutative operator algebra, is captured by its commutative subalgebras. We briefly survey this programme, and zoom in one specific incarnation: any bilinear associative function on the set of n-by-n matrices over a field of characteristic not two, that makes the same vectors orthogonal as ordinary matrix multiplication and gives the same trace as ordinary matrix multiplication, must in fact be ordinary matrix multiplication (or its opposite). Model-theoretic questions about the hypotheses and scope of this theorem are raised.

How to attach non-image files to pages, and how to insert links for those files into your content.
Wed, 11 Feb 2015

11:00 - 12:30
N3.12

The Poincaré conjecture in dimensions 3 and 4.

Alejandro Betancourt
(Oxford)
Abstract

In this talk we will review some of the main ideas around Hamilton's program for the Ricci flow and see how they fit together to provide a proof of the Poincaré conjecture in dimension 3. We will then analyse this tools in the context of 4-manifolds.

Tue, 24 Feb 2015

14:30 - 15:00
L5

A Cell Based Particle Method for Modelling Dynamic Interfaces

Sean Hon
(University of Oxford)
Abstract
We propose several modifications to the grid based particle method (GBPM) for moving interface modelling. There are several nice features of the proposed algorithm. The new method can significantly improve the distribution of sampling particles on the evolving interface. Unlike the original GBPM where footpoints (sampling points) tend to cluster to each other, the sampling points in the new method tend to be better separated on the interface. Moreover, by replacing the grid-based discretisation using the cell-based discretisation, we naturally decompose the interface into segments so that we can easily approximate surface integrals. As a possible alternative to the local polynomial least square approximation, we also study a geometric basis for local reconstruction in the resampling step. We will show that such modification can simplify the overall implementations. Numerical examples in two- and three-dimensions will show that the algorithm is computationally efficient and accurate.
Mon, 09 Feb 2015
15:45
C6

The symmetries of the free factor complex

Martin Bridson
(Oxford)
Abstract

I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a
free group $F$ is exactly $Aut(F)$, provided that $F$ has rank at least $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.

Tue, 17 Feb 2015

14:30 - 15:00
L5

All-at-once solution of time-dependant PDE-constrained optimization problems

Eleanor McDonald
(University of Oxford)
Abstract

All-at-once schemes aim to solve all time-steps of parabolic PDE-constrained optimization problems in one coupled computation, leading to exceedingly large linear systems requiring efficient iterative methods. We present a new block diagonal preconditioner which is both optimal with respect to the mesh parameter and parallelizable over time, thus can provide significant speed- up. We will present numerical results to demonstrate the effectiveness of this preconditioner.

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