Beyond Part III

Young Researchers in Mathematics 2009

16-18 April 2009

Centre for Mathematical Sciences, Cambridge

Applied & Computational Analysis

Talks will be in the Isaac Newton Institute Gatehouse.



Observation Subset Selection - Rachael Tappenden

In many applications involving image reconstruction, signal observation time is limited. This emphasizes the requirement for optimal observation selection algorithms. A selection criterion using the trace of a matrix forms the basis of two existing algorithms, the Sequential Backward Selection and Sequential Forward Selection algorithms. Neither is optimal although both generally perform well. Here we introduce a trace row-exchange criterion to further improve the quality of the selected subset and introduce another observation selection criterion based upon the determinant of a matrix.

Clifford Analysis and Boundary Value Problems - Anthony Ashton

We examine the so called "Clifford Analysis", which extends classical results of complex analysis to higher dimensions, using standard results about Clifford algebras. We then look at some classical boundary value problems from a Clifford analysis perspective. No knowledge of Clifford algebras will be assumed.

Difference Conservation Laws - a math approach (accessible talk) - Timothy Grant

Some of the most fundamental properties of PDEs are expressed as conservation laws; the same is true for partial difference equations. A fundamental question is the following: when are two conservation laws essentially different? For PDEs the answer is found by calculating a function called the characteristic however, until now, no such characteristic has been known for partial difference equations. This talk describes the problem, our search for a characteristic and the surprisingly simple solution. The talk will be accessible to a general applied maths audience; no specialist knowledge is assumed.

Bifurcation theory for localised states - Jonathan Dawes (keynote speaker)

Much of our understanding of nonlinear systems revolves around what the system 'settles down to' at long times, and how this changes qualitatively as we vary control parameters. The payoff for understanding the generic qualitative changes ('bifurcations') that occur is that often the same bifurcations appear in mathematical models and experiments with very different physics.
I will illustrate this idea by discussing the large collection of nonlinear dissipative systems (for example, from fluid mechanics) which, when subjected to uniform external forcing, produce (counterintuitively) spatially-localised responses. In the simplest case, the set of localised equilibrium states (and they're not solitons in the classical sense) has a nice structure and a convincing theoretical explanation, but it's also clear that this is only the tip of the iceberg.

Discrete Snaking: Localized States in Nonlinear Lattices - Chris Taylor

A common technique when analyzing the behaviour of a spatially distributed system is to take the 'continuum limit', averaging over the discrete microstructure to form a PDE model for the system. When is this technique valid, and when does it gloss over vital features of the underlying discrete problem? We will explore possible answers to this question using techniques from dynamical systems and bifurcation theory, focusing in particular on localized responses of the system to a spatially uniform forcing – a very 'discrete' property indeed.

Collective motion of swimming microorganisms - a math approach - Alexander Lorz

We consider coupled chemotaxis-fluid models aimed to describe swimming bacteria, which are able exert bio-convective flow patterns on length scales much larger than the bacteria size. When looking at the microorganisms in detail, experimentalists have observed that high concentration of swimming microorganisms show collective behaviour, in the sense that each organism aligns its swimming direction with those of its neighbours. We discuss different PDE models for the phenomena, give results and challenges in analysing them.

A New Approach to Boundary Value Problems: The Fluid Loaded Plate - Anthony Ashton

We look at a classical problem from hydrodynamics using a new, unified approach to boundary value problems. The fully non-linear problem is first considered, after which we look at the initial-boundary value problem associated with the linear equations on both the full and half line, and establish well-posedness.