Workshop Reports 2005

This page holds short reports on ICMS workshops from June 2005 onwards. Each short report is followed by links to the pdf file of a more detailed report and the workshop's original webpages where you will find the timetable and list of participants. 

Probabilistic Limit Laws for Dynamical Systems 13 -17 June

For the simplest chaotic (but deterministic) dynamical systems, it is often the case that ``loss of memory'' or decay of correlations occurs exponentially quickly  so that time-series measurements are only weakly dependent and obey the laws encountered in elementary probability such as the strong law of large numbers and the central limit theorem.

In recent years, it has been recognised that for the kinds of dynamical systems that typically arise in applications often either the elementary probability techniques fail, or the statistical laws satisfied by the system are not standard (slow decay of correlations, non-central limit theorem behaviour), or both. An important programme has been to understand such systems.

By importing and developing a wide range of techniques from probability theory into the study of dynamical systems, the class of systems for which we have a good understanding of statistical properties has broadened dramatically in the last ten years. Many of these recent insights were the work of conference participants, and the workshop was an opportunity to share their latest results and techniques.  The ensuing discussions generated many new and significant ideas for progress in this area of research.
Workshop timetable and participants | Detailed Report (pdf file)

Algebraic K- and L-theory of Infinite Groups 27 June-1 July

Manifolds are a large-scale generalization of standard Euclidean space, and the most important topological spaces for applications to geometry and physics. The topology of manifolds is closely related to the algebraic properties of modules and quadratic forms over the fundamental group rings. Current research is particularly concerned with infinite groups, where there is a deep interplay between the geometry and algebra. Many of the striking advances made in this field have come from the application of geometric methods to algebraic problems. Geometric group theory has uncovered fascinating new phenomena in the world of infinite discrete groups. On the other hand, the central conjectures of Borel, Novikov and Farrell-Jones have only been verified for groups which connect fairly closely to non-positively curved geometries. The purpose of the meeting was to bring together leading researchers in these fields, both to report on their results and to introduce the subject to a younger generation of mathematicians.
Workshop timetable and participants | Detailed Report (pdf file)

Coagulation-Fragmentation Processes: Theory and Application 4-8 July

The workshop dealt with mathematical and modelling aspects of coagulation-fragmentation processes. Such processes are of importance in describing phenomena in many areas of natural sciences, for example in controlled manufacturing of nanoparticles, molecular beam epitaxy and at the same time in the formation of planetoids. Mathematically, these processes are described by systems of ordinary differential or integro-differential equations. They pose deep analytical and numerical questions and constitute an active area of research. In this area there is a wide scope for interaction between analysts working on both deterministic and stochastic models, numerical analysts, modellers and experimentalists. All these specialisations were well represented at the workshop. A variety of presentations were made at the meeting; fifteen invited speakers gave hour-long talks, and a further twelve shorter contributed talks were also given. As well as providing a forum where experts from the same area reported their most recent results, several talks served to introduce delegates to new methods of analysis and areas of application.
Workshop timetable and participants | Detailed Report (pdf file)

Optimal Transportation, Transport Equations and Hydrodynamics 11-17 July

Optimal Transportation Theory comprises the recent developments related to the classical Monge mass transfer problem, where a mapping is sought that transforms one prescribed finite measureinto another, while minimising a specified cost functional. This subject derived a great impetus from Brenier's work of the late 1980s providing a monotone mapping solving the Monge problem with a quadratic cost, and the subsequent Caffarelli regularity theory. The so-called Brenier map (styled map with convex potential by Caffarelli) has acquired a wide range of applications to geometric inequalities and optimisation problems, and many of the latest developments were discussed at the workshop.

Transport equations describe how a scalar field evolves under the flow of a vector field; the Di Perna-Lions theory of the late 1980s showed how to deal with non-smooth vector fields having Sobolev regularity by introducing the notion of renormalised solution, and recent work of Ambrosio has accommodated vector fields of bounded variation, using a different method derived from geometric measure theory. The later result had been expected for many years and has a wide range of applications, as well as deep connections with the theory of multidimensional systems of hyperbolic conservation laws, one of the most challenging field in nonlinear PDEs.

The vorticity equation of planar hydrodynamics comprises a transport equation whose vector field is the fluid velocity, coupled with the Biot-Savart law to calculate the velocity in terms of the vorticity. In recent years Cullen and his collaborators have been developing a mathematical theory for the semi-geostrophic model of ocean and atmosphere dynamics, which has a formulation expressing the dynamics in terms of an optimal transportation problem, and a dual formulation involving a transport equation by a vector field of bounded variation, so that it impinges on all three themes of the meeting.
Workshop timetable and participants | Detailed Report (pdf file)

Workshop on Dynamical Problems in Mathematical Materials Science
18 - 23 July 

Over the past 20 years, mathematical materials science has become a vibrant discipline, with a very significant impact both on mathematics
and materials science:
  • On the one hand, materials science provides a very rich source of problems in various areas of mathematics. For example, the link between the microscopic structure and macroscopic properties can be explored by certain mathematical techniques, namely homogenisation and  relaxation. Over the past few decades this has led to substantial progress in both numerical and mathematical analysis. Mathematical progress, especially in the context of the calculus of variations, has produced a good understanding of several static problems.
  • On the other hand, the field of materials science has been and continues to be revolutionised by the availability of experimental and theoretical advances in multiscale methods based on advanced mathematics. The newly gained ability to design materials with intricate features on an almost atomistic level calls for a theoretical foundation.
The challenge is to understand the dynamics of multiscale problems in materials science, and how dynamics on one scale affect those on the other. Activities in this field are now just emerging and were the theme of this conference.
Workshop timetable and participants | Detailed Report (pdf file)

Workshop on Parameter Estimation in Continuous Time Models
5 - 9 December

Continuous time models arise very naturally in the modelling of a variety of phenomena where random fluctuations are present.  The models covered by the workshop fell broadly into two catagories, those driven by stochastic differential equations (SDEs) and those described by point processes. Applications for both types of model come from diverse fields such as econometrics and finance, molecular  modelling and the atmosphere/ocean scinces and modell int the spread of infectious diseases. As part of the overall modelling strategy in such application domains, it is frequently desirable to fit parameters driving the continuous time dynamics to the data, in order to optimize predictive capability. As the generation and gathering of data in all application domains has gathered pace over the last decade, and as computational power has increased similarly, so the possibility of making reliable estimates of parameters had increased.

The  subject has been developing in may exciting ways in many different fields, both methodological and application specific, and the workshop brought together researchers engaged in different aspects of the field.  The primary importance of the workshop was in exposing researchers from a wide variety of fields to the work of one another, together with the resulting identification of important areas for future research.Workshop timetable and participants | Detailed Report (pdf file)

Workshop on Rogue Waves
12 - 15 December

Rogue, or freak, waves is currently a very hot topic. At the same time, it is a topic of substance in nonlinear wave theory, and an ICMS workshop was timely and appropriate. Briefly, a rogue wave is the rare transient occurrence of a wave whose amplitude is significantly larger than the background sea-state. A commonly-used ad hoc definition is a wave that is at least 2.2 larger then the significant wave height. Although they are rare events, just how rare is not clear; a spate of recent ocean observations suggest they are not as rare as had been thought. These destructive waves are of major concern for shipping and off-shore engineering.  Based on various numerical and analytical models, several dynamical mechanisms have been proposed for their occurrence; these included Fourier superposition of many small waves with suitable phase relations, nonlinear focussing of wave energy, and wave refraction by currents and/or topography. However, a detailed and definitive understanding of rogue waves, and related phenomena, is not presently available.  Hence, it was felt that was a need for an ICMS workshop to look at rogue waves from a more mathematical and fundamental attitude.
Workshop timetable and participants | Detailed Report (pdf file)
Links to abstracts, papers and presentations as available for each talk
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