MATHEMATICS APPLIED TO PHYSICAL PROBLEMS AT THE NEWTON INSTITUTE
Scientist in charge::
Professor H K Moffatt
Isaac Newton Institute
20 Clarkson Road
|Type of event: |
The figure illustrates the interaction of a one Jupiter mass protoplanet with a standard protoplanetary disc. The time measured in orbital periods after introduction of the protoplanet is indicated in the top right hand corner of each panel. During the early evolution spiral density waves are excited and a gap forms. Subsequently, accretion of the inner disc leaves a large cavity while gravitational interaction with the outer disc causes the protoplanet to migrate inwards. Such inward migration may account for the presence of extrasolar gas giant planets close to their central star.
| Summary of the event |
Arithmetic Geometry (23 March to 3 April 1998)
Diophantine equations are polynomial equations with integer coefficients for which one seeks to find solutions that are integers or rational numbers. This is a topic of central importance in Number Theory (ie Arithmetic), whose origins can be traced back to Babylonian times.
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:
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