Warwick Symposium on Ergodic Theory and Dynamical Systems (ETDS) 2010-2011

This is a proposal for a symposium on to be held during the academic year 2010 - 2011 at the the Mathematics Research Center (MRC), of the University of Warwick. The overall theme of the Symposium is Ergodic Theory and Dynamical Systems , two closely coupled scientific fields within Pure mathematics which are at the forefront of modern scientific research. Broadly speaking, they both deal with the behaviour of orbits of points under iterative applications of transformations on a suitable space, and diverse applications of these basic principles to the global behaviour of the evolution of systems. One might say that Dynamical Systems deals with more global topological aspects and Ergodic Theory deals more with measure theoretic aspects (e.g., typical points). The intimate interaction between these fields is encapsulated in their pairing in the title of the leading international journal Ergodic Theory and Dynamical Systems , based in the Department of Mathematics, at Warwick University.Compared with other fields of mathematics, both Ergodic Theory and Dynamical Systems are relatively young (with perhaps merely a century of development). However, they have not only developed into exciting and active areas in their own right, but they have also provided effective tools in other areas of mathematics - often providing the key to stunning resolutions of long standing conjectures in seeming different fields.This is exemplified by applications of Ergodic Theory to Number Theory, where Margulis' solution of the Oppenheim Conjecture; Furstenberg's proof of the Szemerdi Conjecture; and the Einseidler-Katok-Lindenstrauss approach to the Littlewood Conjecture have all been landmarks in the rapid advancement of this approach.Within the UK, we are fortunate to have a number of individual centres of excellence for both Ergodic Theory and Dynamical Systems, among which Warwick has the longest tradition, and remains a pioneer. Indeed, many of the national leaders in this field were trained at Warwick. For these reasons, Warwick is a natural home to such a symposium - not withstanding Warwick's tradition and experience in hosting symposia over many years.The aim of this symposium is two fold. Firstly to make the greatest possible scientific contribution to the field, drawing upon the talents of both UK and overseas experts of the highest calibre. Secondly, to use this as an opportunity to draw upon the very considerable native talent within this country to support the development of this field in the UK in a unified and coherent way. By promoting greater interaction between the UK research groups we hope to develop a movement which is greater than the sum of its individual parts In practical terms, there will be six constituent workshops, to which will be invited leading international and national experts. To make the symposium as inclusive as possible, and to make the impact as broad as possible, the Warwick based organisers will be assisted by a number of specialist coorganisers for each of the workshops. In addition, there will be a visitor programme and a number of other supporting activities designed to sustain the level of activity throughout the year.The organizers of the symposium (and the Investigators of this proposal) are Mark Pollicott and Sebastian van Strien, both professors at Warwick. They are committed to making this Symposium as effective as possible in supporting and promoting cutting edge research (both nationally and internationally) in Ergodic Theory and Dynamical Systems and the broader mathematical community.

Although the primary beneficiaries of this project will be researchers working in mathematics, in general, and pure dynamical systems, in particular, we envisage that this project will eventually have impact in a broader setting. However, this project is not aiming to explore these applications, but has as its immediate aim to: (i) strengthen the knowledge base within the field of dynamical systems (and more generally mathematics) in the UK, maintaining leadership; (ii) to bring the best people wordwide to this symposium in order to enhance the UKs research basis; (iii) to be at the cutting edge of research in a active and vibrant area. In particular, this project will support UK universities' drive to be among the prime generators of top-quality research. Of course this is not only a matter of prestige, as no country in the world can expect to maintain long term economic success without a strong strong scientific base and strong universities and, in particular, without a strong mathematical foundation. The nature of pure mathematical research is that it is traditionally somewhat removed from direct commercial and govenment usage. However, we might expect that the quality and cutting edge nature of this proposal will ultimately have far wider implications which are harder to predict at the moment. We would expect there to be a significant impact, but only over the longer term. An example of the interface of the interaction between Pure Mathematics and a government agency is via the Heilbronn Institute in Bristol. Although we don't plan any direct interaction with this Institute, we anticipate that there could be interactions from visitors with shared interests, particularly in number theory and algebra. During the December workshop we intend to organise a small exhibition with illustrations of mathematical computer experiments. These illustrations (showing fractal-like images), are not only aesthetically pleasing, but also can give real insight to both lay-people and experts into mathematical discoveries. By organising this exhibition, we expect that we will be able to help enhance the interest in mathematics amongst the general public (and in particular amongst potential mathematics students) and make a small contribution to avoiding the crisis of interest in mathematics which has developed in many European countries. The workshops will have survey talks, but also mini-courses which will an ideal way to learn a topic in quite some depth. We aim to publish notes of these survey talks and of the mini-courses (provided the lecturers will be willing to coorporate with this). We envisage that there will be a number of keynote talks by leading specialists aimed at a broader audience. This will be part of a concerted effort to broaden the impact of the symposium. We also have the provision of using using our dedicated Access Grid Node to both record and distribute talks of broader interests to a wider audience. This method has already proved successful with the EPSRC-funded TCC (Taught Course Centre), also involving Bath, Bristol, Imperial and Oxford. We anticipate that participants at the symposium will also travel to other institutions to give talks and/or engage in collaboration. This Symposium will not only be about cutting edge research, but also about setting the research agenda in this field until 2020. The structure of this Symposium has been carefully designed to stimulate debate and discussion on fruitful future research directions. We will make every effort to ensure that there is a fruitful and productive interaction between individuals with a common interest in ergodic theory and dynamical systems, in the broad sense, but with different backgrounds and perspectives. We are confident that this will be a very effective way to generate new ideas and avenues for research.