Workshop Reports 2006

This page holds short reports on ICMS workshops during 2006. 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, a list of participants and in some cases abstracts or presentations.

3-manifolds after Perelman 13-17 March

The goal of 3-dimensional topology is to understand all mathematically possible ”universes”, i.e. spaces (3-manifolds) that are locally like ordinary 3-dimensional Euclidean space but whose global structure might be quite complicated. Since the mid 1970’s, the guiding light of the subject has been Thurston’s Geometrization Conjecture, which asserts that every 3-manifold is built up from pieces that have a geometric (as opposed to a merely topological) structure. In 2003 a proof was announced by Perelman, using analytic techniques.

The aim of the workshop was to have leading researchers describe the current developments in 3-dimensional topology that are more or less independent of Perelman’s work, and what they see as the important directions and problems for the future, now that the Geometrization Conjecture has been proved. The main focus of the program was four series of three lectures given by four leading researchers, who have been responsible for some of the most exciting recent developments in the subject: Jeffrey Brock, Marc Lackenby, Peter Ozsvath and Peter Shalen. There were an additional 11 1-hour lectures given by other internationally recognized experts. The meeting also provided a useful forum for discussion among the participants. We summarize some of the mathematical content below.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Mathematical Population Genetics 26-30 March

In order to correctly interpret the huge mass of molecular data and make inferences about our evolutionary past we must find ways to disentangle and distinguish the distinct causes of genetic variation. This raises a plethora of mathematical problems. First evolutionary processes must be incorporated into tractable models, then robust statistical tests must be devised that can distinguish between, for example, the effects of natural selection and those of demography, the effects of fluctuating selection and those of balancing selection and so on. To achieve this programme it is vital to bring together mathematicians and biologists to highlight the techniques available, identify some interesting biological problems and foster lasting collaborations between the two communities and this was what motivated this meeting.

The speakers made strenuous efforts to make their presentations accessible to the very heterogeneous audience (about half from biology departments and half from mathematics departments) and where available the electronic versions of their presentations are linked into the programme below.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Quantile regression, LMS method and robust statistics in the 21st century 19-23 June

Regression analysis is one of the most popular and powerful techniques in statistics. Quantile regressions, with median regression as a special case, are particularly useful for extracting essential features, and finding structures and relations in complex datasets. Quantile regression has now been used in almost all scientific fields, including potential good applications in two current popular areas: Finance and Bioinformatics. The recent flourishing of publications and research interest in quantile regression, LMS methods and robust regression is a timely reminder that these methods are evolving into mainstream statistical techniques for the analysis of data from today's vast range of application areas.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Applied Asymptotics and Modelling 26-30 June

A key aim of the meeting was to bring together a significant number of different groups of very active, yet often disconnected (geographically and ideologically) workers both in exponential asymptotics and beyond. A meeting covering such a broad range of topics and applications had not been attempted for a decade in the UK, since when the state of the subject has advanced dramatically. To that end we planned the meeting around themed days.

  • Monday: Recent developments in exponential asymptotics for PDEs; zeta functions.
  • Tuesday: New challenges in asymptotic applications I: biological and stochastic systems to the onset of turbulence through coupled exponential scales.
  • Wednesday: Applications of Riemann Hilbert methods.
  • Thursday: New challenges in asymptotic applications II: from field theory to nanotechnology and elasticity.
  • Friday: Asymptotics applications to dynamical systems, Painlev´e analysis. Recent significant advances in integral asymptotics.

The meeting attracted over 40 researchers worldwide, spanning pure and applied mathematics, and from physics to engineering. The informal atmosphere of ICMS and the facilities of the surrounding town were ideal for new contacts that were forged through planned and spontaneous discussion sessions which has resulted in several new concrete collaborations.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

New directions in applied probability: stochastic networks and beyond 10-14 July

This workshop, made possible by a grant from EPSRC and ICMS, dealt with the latest trends and directions in applied probability with emphasis on applications to stochastic networks and related stochastic systems.

The main goal was the discussion of mathematics behind a variety of applications and, to achieve this, we invited a number of researchers, beyond the traditional area known as “Stochastic Networks.” In the workshop, we explored relations between different sub-areas of Theoretical and Applied Probability and how these can be used to boost further directions in Stochastic Networks.

One of the aims of the workshop was to strengthen UK involvement in this rapidly developing field that has several important industrial applications, e.g. in communications, in transportation, in the power industry, and in biology.

The outcomes of the workshop were a deep exchange of ideas. Several of our colleagues have already pointed out to us that such workshops are what we ought to have more frequently, for they had the opportunity to discuss things beyond the ones they are usually involved with.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Extreme Kähler metrics and stability 17-21 July

In mathematics many of the most striking problems and their solutions lie at the interface of apparently separate disciplines. This workshop brought together researchers in the separate, but related, fields of algebraic geometry, geometric analysis, symplectic geometry and differential geometry to discuss a very difficult problem at the forefront of current research. This problem is not so easy to describe in general terms. In rough terms, the extremal Kähler metrics of the title are generalizations of metrics of constant Gauss curvature on ordinary 2-dimensional surfaces. From the point of view of analysis, they satisfy a fully nonlinear fourth- order partial differential equation, and for this type of equation no general methods are currently available. However, surfaces can also be viewed as complex algebraic curves, in other words given by polynomial equations in two variables. It is from this viewpoint that the notion of K-stability is defined.

The main theme of the workshop was to develop the conjectural relationship between the existence of extremal Kähler metrics on the one hand, and K-stability of the underlying complex variety (higher-dimensional version of algebraic curve) on the other. This is an extremely challenging problem because of the disparate nature of the two objects being compared. While this problem remains open, the workshop gave experts in the relevant areas an opportunity to report on simplified versions of this problem where partial results have been obtained, as well as appropriate background material both from analysis and algebraic geometry. New collaborations were started during the workshop and the feedback from participants was uniformly very positive. The mix of mathematicians with different skills, focusing on a specific hard problem, has worked extremely well.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Mathematical models of development and learning in the nervous system 21-22 July

The formation and modification of connections between nerve cells is of critical importance in understanding the complex and robust computations of the nervous system. Experimental neuroscience provides large quantities of data on the scale of neurons, synapses and molecules as well as on the larger scale of regional brain activity and behaviour. Mathematical models have an important role in formalising the key rules of development and learning that tie together these levels.

A wide variety of mathematical and computational formalisms and techniques were discussed. Although the application of the ideas and methods is more general, some particular neurobiological issues formed the context of the modelling, including:

  • the dynamics of how neurons are assembled from proteins and how they follow very noisy molecular gradients to find their target regions;
  • the origin of topographic mappings of neurons in one structure (e.g. the retina) to another (e.g. the optic tectum) by molecular gradient matching;
  • how the brain can maintain stable memories in the face of constantly changing the representations;
  • how cells come to be laid out in semi-regular mosaics in structures such as the retina;
  • the role of neuronal activity in self-organisation of neuronal connectivity; and
  • how the neurotransmitter dopamine affects learning by gating the plasticity of synapses.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Mini-programme on Algebraic theory of differential equations 31 July - 11 August

This workshop covered Differential Algebra, Differential Galois Theory, their model-theoretic aspects and the theory of integrable differential and difference systems, areas in which, in different ways, algebraic methods are applied to differential equations. It aimed at the promotion of interdisciplinary research, so that these approaches could complement and inform one another.

Differential Galois theory works in direct analogy with usual Galois theory. Model theory has recently shown, for example, that one can always in principle compute the differential Galois group of a linear differential equation. Algebraic structures are in the heart of the theory of integrable systems. Study of Lax pairs and their reductions has led to the characterization of infinite dimensional automorphic Lie algebras, a potentially important class of algebraic structures. Non-local extensions of differential fields are required for the development of the theory of multi-dimensional integrable systems. Symmetry groups and algebras appear in several approaches, their applications together with new effective computational methods have been developed. Variational equations and isomonodromy provide one of the bridges to the differential Galois theory. All these were brought together in the workshop.

The workshop was preceded by a week-long more pedagogical school, overlapping substantially both in the areas covered and in the participants. The workshop consisted of individual talks of a more advanced character bringing participants to the frontiers of research.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Metric entropy and applications in analysis, learning theory and probability 11-15 September

The main aim of the workshop was to bring together active mathematicians from three different areas - analysis, learning theory, and probability theory - the link being their common interest in metric entropy. The concept of metric entropy plays a prominent role and has interesting applications in all three areas, but so far there has not been very much co-operation between the different research directions. The workshop provided a forum to intensify the existing and to establish new contacts, giving all participants the opportunity to learn from each other, to exchange ideas, new results and techniques, to discuss open problems and promising directions of future research. The core of the workshop were the survey lectures on the use of entropy in analysis (theory of function spaces, interpolation theory), learning theory (statistical learning theory, machine learning) and probability (small deviation problems). Some other main talks were devoted to important open problems connected with metric entropy (interpolation of compactness by the complex method, duality of entropy numbers), or relevant new developments in the corresponding fields. In the short contributions recent results on more specific topics were presented.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

Credit risk under Lévy models 19-21 September

The goal of the workshop was to explore the marriage between the probabilistic and analytical study of Lévy processes and the recent interest in credit risk models. The workshop addressed the apparent potential for theoretical and applied results, innovative and accurate model structure, and exposition of new phenomena in this new interdisciplinary intersection.

Specific topics addressed included the following which sit at the interface between financial and insurance mathematics:

  • Fundamental credit risk models taking into account jumps and their mathematical consequences.
  • Passage problems and optimal default.
  • Distributional aspects of integrated exponential Lévy processes and applications.
  • The use of integro-differential equations, boundary value problems, theoretical and numerical solutions thereof.
  • Multivariate asset modeling using Lévy processes in credit models.
  • Valuation and hedging of credit derivatives in the presence of jumps.

Our aim was to create an environment from which researchers could become aware of techniques and allow for interaction to the extent that new collaborations could be formed. Concluding the workshop with a round the table discussion proved to be a very effective way of establishing a sense of direction in this applied field of mathematics which unavoidably needs some guidance from the finance industry itself. In addition, it was important to the workshop that there would be a significant presence from younger researchers.

At the end of the workshop there was a one day conference (self funded and held at the Royal Society of Edinburgh) which took the themes of the ICMS workshop to a more industrial audience as well as taking advantage of the presence of the senior researchers in the field of financial mathematics from the ICMS workshop.

Workshop timetable and participants | Detailed Report (pdf file) | Return to top of page

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