MATHEMATICS APPLIED TO PHYSICAL PROBLEMS AT THE NEWTON INSTITUTE

ASTROnewton970203/EW/ac

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01223 335980

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hkm2@newton.cam.ac.uk

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University of Cambridge

Contract N°:
ERBFMMACT970203

Number theorists have made important progress on certain special classes of equations, most notably those describing curves: f (x,y) = 0. Various conjectures predict a deep relationship between the distribution of rational solutions and the geometry of the set of all complex solutions. This geometric perspective on arithmetic problems provided the central theme of the 6-month programme on Arithmetic Geometry held at the Newton Institute (January to July 1998). The EC Summer School, held within this programme, sought to disseminate recent developments in this field, which involves creative interaction between number theory, algebraic geometry, analysis and algebraic topology, subjects at the very core of modern mathematics.

Part I: Instructional Conference, Current Trends in Arithmetical Algebraic Geometry

The goal of this instructional conference was to bring the audience of predominantly young researchers (postdocs and PhD students) in direct contact with some of the most important current research. There were four lecturers, A Goncharov (Polylogarithms, regulators, values of L-functions), M Nakamaye (Diophantine approximation on algebraic varieties), P Salberger (Manin's conjecture on points of bounded height) and V Voevodsky (Motivic homotopy theory), each giving 4-5 one-hour lectures. While these talks presented very advanced and recent material, the speakers succeeded very well in presenting it with enough details and preliminaries, so that the large and uneven audience would catch most of what was said.

There were about 80 participants, from Austria, Bulgaria, Czech Republic, France, Germany, India, Italy, Netherlands, Russia, Spain, Sweden, UK and USA.

The organisers encouraged all participants to ask questions, and, indeed, there were many, both during the lectures and at the end of them. Notes had been written by the speakers prior to the meeting, and they got widely distributed.

Part II: Rational Points

This was a combination of a research and a survey conference, organised by J-L Colliot-Th‚lŠne and P Swinnerton-Dyer. The general aim of the conference was to give the participants a global view of what is known and what is to be expected of rational points, depending on the place of varieties in the geometric classification. There were about 100 participants, from Austria, Belarus, Bulgaria, Croatia, Czech Republic, France, Germany, Hungary, India, Israel, Italy, Netherlands, Russia, Spain, Sweden, Switzerland, UK and USA. The lectures were given by D Abramovich, F Beukers, J-B Bost, A Buium, A Corti, G Faltings, D Harari, R Heath-Brown, L Merel, M MacQuillan, J-F Mestre, E Peyre, B Poonen, N Shepherd-Barron, J Silverman, A Skorobogatov, M Strauch, P Swinnerton-Dyer, Yu Tschinkel and P Vojta.

Astrophysical Discs (22 to 27 June 1998)

Many astrophysical systems consist of matter organised in differentially rotating discs, in which gravitational and centrifugal forces are in approximate equilibrium. These systems may be dominated by a central gravitating mass; the angular momentum of the disc inhibits its collapse onto this central mass.

Familiar examples are planetary discs (eg the rings of Saturn); gaseous discs around proto-stellar objects, the disc being a product of the star-formation process; and galactic discs in which matter spirals inwards to fuel massive black holes located at galactic centres.

The evolution of such astrophysical discs is controlled by complex processes of transport of mass and angular momentum through the fluid medium of which the disc is composed. Magnetic fields associated with currents in the disc are a serious but unavoidable complication. Many of the processes observed in such systems, eg the production of outflows and jets from the discs, warping of the disc (see diagram) and interactions with orbiting companion stars, are subjects of very active current research.

This Summer School was held at the end of a six-month programme on the Dynamics of Astrophysical Discs at the Newton Institute in which all of these topics had been intensively studied. The main object was to provide background material covering all of the areas of astrophysics, ranging from the solar system where discs around planets occur, to the scale of galaxies where discs of orbiting stars occur. A priority was to make this material accessible to young researchers embarking on their first work in the field while also disseminating recent results and discussion of work in progress. The lecturers were asked to combine material they had been working on during the programme with a general review of the topic they were lecturing on. Overall a summary of the work undertaken during the programme would then appear in the context of the background of the field.

Each topic was allocated a major reviewer who lectured for one hour, together with one or two other speakers who talked for twenty minutes, concentrating more on recent developments and work undertaken during the programme. Proceedings from the summer school, contributed to by all of the speakers, are due to be published in the Publications of the Astronomical Society of the Pacific Conference Series edited by two of the principal organisers, J Goodman and J Sellwood.

The topics covered included: Magnetohydrodynamic turbulence and the origin of disc viscosity, discs in active galactic nuclei, advection dominated discs, mechanisms for producing warped discs which are known to occur in the case of active galactic nuclei and protostellar discs and also in many galactic discs, observed properties of discs in cataclysmic and X ray binaries, eruptive behaviour and thermal instability in dwarf nova discs, jets in protostellar discs, jets in active galactic nuclei, collisionless stellar discs, the theory of bar instabilities in galactic discs, the origin of planetary systems in protostellar discs and disc companion interactions including those resulting from galaxy collisions and an embedded orbiting satellite.

The lecturers selected had recently produced state-of-the-art results in their area and the students had presentations on these in all cases. The standard of the lectures was very high and lecturers took the time out to receive questions from and interact with the students in all cases. In this respect the school was a great success.

The school attracted 27 students and postdoctoral fellows supported by the EC and other young scientists supported from sources such as EPSRC and the London Mathematical Society. They were invited to present posters on their recent research and many did this. This added to the already lively and stimulating environment. Apart from concentrating on material they had received in the lectures, the students were able to meet with and discuss their own work with the most eminent people in the field. This was also an important element contributing to the success of the summer school.

In summary, the students had as wide ranging presentation of the field of discs in astrophysics as could be obtained anywhere. In addition to receiving excellent review material from expert and active workers in the field, they were presented with the most recent results. They were also able to discuss their own work with the lecturers. Finally a comprehensive volume of the proceedings edited by two of the principal organisers should become available in 1999.

Bayesian Signal Processing (20 to 31 July 1998)

This workshop on Bayesian Signal Processing was the star of a six-month programme held at the Isaac Newton Institute on Nonlinear and Nonstationary Signal Processing.

The reason for holding such a six-month programme was the fact that the classical theory of signal processing is based on models which are stationary, linear and in many cases also Gaussian. Recent advances in time series and the theory of signal processing have drawn attention to many new models and methods. Among these are nonlinear autoregressive and nonlinear state-space models, state-space models with time-varying or state-dependent coefficients as models for nonstationary and nonlinear series, linear non-Gaussian processes which pose some specific problems not encountered with Gaussian processes, methods derived from the theory of dynamical systems, and many others. Most of the methods for the analysis of such time sequences (e.g. for detection of long-term trends, for the prediction of extreme values, or for the extraction of structure from an apparently chaotic signal) originated in engineering, but are now applied to other areas such as financial time series, the environmental sciences, physiology, etc.

Many signals of current interest can be modelled in terms of nonlinear systems which gave rise to the observed data. These nonlinear systems can, under certain conditions, give rise to 'chaos'. Dynamic models are fundamental to most branches of signal processing and communications, occurring for example in the analysis of speech and music, in the modelling of wireless communication channels and in radar tracking problems. Realistic dynamic models in these kinds of applications involve non-linear and non-Gaussian structure and are therefore not amenable to analysis without resort to numerical approximations. In modern times, the gold standard for numerical approximation in Bayesian analyses is via iterative simulation methods or MCMC. Unfortunately, such methods are not naturally embedded or applicable in a context of sequential, on-line analysis and updating in real time; computational efficiency aside, there exist outstanding conceptual and technical challenges to adapting simulation models to such cont exts. These issues were addressed during our programme. Applications and examples are the following:

• The real-time analysis of degraded audio signals, for example old 78rpm records with scratches etc;
• The analysis of EEG signals concerning both the depth of awareness during anaesthesia and the prediction of epileptic attacks;
• Other applications concern the ability to detect signals in a mobile communications environment where the communications channel suffers from fading and time variation, the real-time detection of change-points applied to ion-channel dynamics in pharmacology as well as applications in speech processing. In many instances, methods have developed in an ad hoc way, being designed for specific applications rather than fitted into a general framework. In particular, very often similar problems have been examined by different groups of workers with minimal contact between the groups. The purpose of this programme was to bring together statisticians, engineers and other researchers who use signal processing methodology to unify existing methods and to identify areas of research where new methodology is required.

  Bayesian inference is the methodology which underpins most of the above and it was felt that a workshop dedicated to Bayesian methodology would be most appropriate.

  In recent years there has been an explosion of interest in Bayesian statistics. The development of new algorithms for Bayesian computation, together with the ready availability of computing resources, have made feasible statistical methods which until the present decade were limited to small data sets and restricted classes of models. This "revolution" in statistical methodology has affected every area where statistics is applied, including medical statistics, the social sciences, analysis of financial data and the modelling of large environmental systems.

  Signal processing refers to the class of methodologies available for handling data produced sequentially in time. Most of the methods originated in engineering but are now applied in many other areas as well, eg communications, econometrics, geophysics, physiology, image processing. Classical methods of signal processing, such as the Kalman filter, were based on stochastic processes which are linear and Gaussian. One reason for this restriction was that the computations required for such systems are relatively simple and could be readily implemented using the computing resources of the 1960s and 1970s. However such assumptions are inadequate for many modern applications and more flexible models have emerged. Examples include wavelet methods for time-frequency analysis, Kalman filters with time-varying or state-dependent coefficients, new techniques for non-Gaussian processes, and dynamical systems and chaos as mathematical models for time series. However in many instances the new methodology has developed in an ad hoc way, being designed for specific applications rather than fitted into a unifying framework.

  The main focus and thrust of this workshop was the fact that Bayesian methods provide a unifying methodology whereby different kinds of mathematical models may be examined within a common statistical framework. The workshop brought together the statistical and computational expertise of leading statisticians and the modelling expertise of mathematicians and subject matter specialists, with the broad objective of developing new signal processing tools which make efficient use of modern computational resources while combining the most up-to-date research of both groups of specialists.

  Specific topics that were covered included:

  1. Bayesian methods in general and numerical methods in particular,
  2. Nonlinear and nonstationary time series estimation,
  3. Forecasting and changepoint modelling,
  4. Nonlinear signal processing in econometrics and financial time series,
  5. Dynamical systems and statistics,
  6. Environmental applications and spatial data analysis.

  The workshop had around 100 participants from 18 different countries. Many of the talks were aimed at young research workers and this was combined with more advanced 'state of the art' approaches. The younger scientists had the opportunity to show their work during several poster sessions. The workshop was a huge success and has formed the basis for several collaborations.

  Methods for Molecular Phylogenies (10 to 14 August 1998)

  There is a long and productive history of interplay between genetics on the one hand, and statistics on the other. The 'molecular revolution' over the last 15 years, and in particular the impetus provided by the human genome project, has transformed the field to one with an abundance of data and a paucity of relevant statistical models and techniques for the extraction of useful information from this data. As a consequence of recent advances in computational statistics, vast improvements in the quality of statistical analyses of data currently available from genome projects are now possible. These improvements are expected to have a profound impact on the practice of biological research, and in the longer term, on medical diagnostics and preventive medicine.

  This Summer School was held within the Newton Institute programme Biomolecular Function and Evolution in the Context of the Genome Project (July-December 1998),which was concerned with these broad issues; the School was concerned more specifically with the statistical analysis of 'phylogenetic' data, ie of gene sequences sharing a common ancestry.

  For many years, comparative analysis of such gene sequences concentrated on the inference of evolutionary histories. Varying levels of similarity between homologous sequences were analysed according to simple models of evolutionary change to infer the historical relationships amongst, for example, the major mammalian orders diverging after the extinction of the dinosaurs; amongst modern primate species (humans chimpanzees, gorillas, orang utans etc); or even amongst different strains of the HIV-1 virus within individual infected patients.

  More recently (particularly in the last five years), emphasis has also been given to the study of the processes of molecular evolution. These molecular phylogenetic analyses arose as a scientific discipline with the advent of some of the first sequence data in the later 1960s. However, little data was available for many years. Theoretical (ie, mathematical and statistical) work on data analysis methods in the field advanced only very slowly. Many methods were devised in an ad hoc manner by biologists not trained in statistical methods, and consequently the field became controversial, with ill-understood and 'opposing' methodologies being introduced. Only in the past 10 years has a fuller understanding of the statistical properties of phylogenetic inference methods enabled a truly scientific framework for data analysis to be developed. A large proportion of established researchers around the world are still not fully aware of the 'state of the art' in molecular phylogenetics. The organisers of this Summe r School felt it was valuable to devise a course to introduce modern ideas on the major methods of data analysis, the mathematical and statistical foundations of these methods, and, in a practical vein, the use of the major computer programs available for performing these analyses to younger scientists.

  The presence at the Summer School of Prof Walter Fitch (one of the founders of the study of molecular evolution) and Prof Joseph Felsenstein (the world's most influential researcher in methods for evolutionary analysis of molecular sequences over the last 20 years) laid the foundations for a highly attractive course. In addition, Prof Felsenstein is renowned as the organiser of a highly successful annual course in the USA akin to our Summer School, and the opportunity for EC students to attend a similar course was clearly a great attraction.

  The course structure was devised by Prof Felsenstein. All five mornings were devoted to lectures. On the first day, the lectures were devoted to a general introduction on phylogeny methods and a history of research in the field. The second day had three lectures, each an introduction to one of the three major approaches to phylogenetic analysis. The third day had a series of four lectures on the numerical optimisation problems involved in phylogenetics - a very significant topic, due to the unusual statistical nature of phylogenetic inference and the immense search space that must be considered. The fourth morning had lectures on the statistical properties of phylogenetic analyses, a field which has only become well-studied relatively recently and is still widely misunderstood. The final morning was devoted to three lectures describing a broad range of practical uses for estimated phylogenies. The Summer School organisers were delighted that in every case lecturers were amongst the two or three most in fluential workers in their respective fields.

  All five afternoons of the Summer School were devoted to practical classes, teaching the use of all the major computer software packages applicable to molecular phylogenetics. The content of the five sessions was organised by Dr Frank Wright, and detailed consultation with Prof Felsenstein ensured that each day's practical work was accessible assuming only that the lectures to date had been attended. A typical session would involve two or three software demonstrations, each followed by a period when the students could work together and seek further assistance from the demonstrators and other instructors. The first session was very basic and illustrative, with more demonstration than practical work; by the end of the week students were attempting a very broad range of analyses, using a variety of different software packages unaided. The Summer School organisers were pleased to find that one of the senior authors of every one of the major software packages used was available at some stage during the course, to explain and demonstrate their software and answer students' questions. As well as providing the most up-to-date software, the instructors provided example data sets for the students to analyse. Time was also available for students to get expert assistance with the analysis of data sets of their own. In addition, one data set was devised by the course organisers for the students to analyse unassisted, with brief written reports being submitted in an informal competition.

  The Summer School organisers were agreed that the Summer School was a great success. There was much discussion of whether the course could be repeated regularly. Prof Felsenstein, who is very experienced with similar courses in the USA, commented very favourably on all aspects of the Summer School, and in particular the facilities provided by the Newton Institute and the standard of the students. His advice was that, judging by the attendance at this Summer School, it would be feasible to run such a course on an annual basis.

PROGRAMME OF EVENTS