Workshop on function field arithmetic

The purpose of this proposal is to organise a two day workshop on the arithmetic of function fields at Imperial College in June 2011. This will be the first meeting on the arithmetic of function fields held in the UK and one of its main aims is to popularise this highly developed and prestigious area of modern number theory, which is still rapidly evolving, to the UK mathematical community. The arithmetic of function fields centres around the concepts of Drinfeld modules and t-motives. They proved to be an immensely powerful tool in resolving several central conjectures in Pure Mathematics, namely the global and local Langlands conjectures, the Ramanujan-Petersson conjecture, the Jaquet-Langlands correspondence, and the Carayol-Deligne conjecture. On the other hand although their study were originally undertaken in order to make advance on the above-mentioned conjectures, now it is a discipline on its own, an extremely rich theory which have a counterpart for every object which is studied by classical number theory, including modular varieties and modular forms, L- and Gamma-functions, Galois representations and Hodge theory, Bernoulli numbers and transcendence results and which often closely mirrors, but sometimes tantalising diverges from the arithmetic of number fields. Moreover very often the state of art in the arithmetic of function fields is far more advanced than in the case of number fields, hence the area still serves its role as an important testing ground for the major conjectures of number theory. The area also has deep connections to representation theory, algebraic and arithmetic geometry, group theory and even combinatorics. It is very natural to expect that many major results in this area are still to be discovered. Some of the leading experts have already accepted to speak and attending Ph. D. students will be exposed to the latest methods spanning a large selection of topics. The other aim of the workshop is to stimulate discussion among its participants and further research in this exciting area.

I expect two main types of impact of the workshop. Immediate educational impact. This will take the form of transfer of knowledge and skills from the experts in the area to the participants of the workshop, in particular to the attending Ph.D. students and postdocs. Arithmetic of function fields is a very active, highly respected area of modern number theory, a fact well-documented by the large number of publication of this area in leading mathematical journals, but which is not very well represented in the UK, so such a workshop would be an ideal way to introduce it to current and future researchers in number theory. Moreover the subject has deep connections to many other areas of pure mathematics, in particular to algebraic geometry, group theory, representation theory, and combinatorics. Thus the methods developed for studying the arithmetic of function fields have independent interest to mathematicians from the areas above and I believe they will find the topic covered by the workshop useful in their own work. This impact will be achieved by advertising the workshop to the widest possible audience with background and interest in the subject via web page, mailing lists (London Number Theory Seminar), personal contacts and other research activities organised by the PI. In addition part of the workshop talks will have of introductory or survey character. Longer term impact (research and applications). In addition I expect the workshop to have a lasting impact in stimulating further research in the area. This impact will be achieved in the following way: some of the leading experts in the subject have been invited. The programme will promote open problems and also encourage discussion among the participants during lunch and coffee breaks and even after the event (via a mail list). Resulting research will be published in leading mathematical journals and made available to free preprint archives like arxiv.org. Since this area serves as an important testing ground for conjectures in classical number theory, I also expect lasting impact on the arithmetic of number fields, too. Long term applications. Even though the research topic of the workshop is purely theoretical its long term impact could benefit all sections of society: better understanding of the arithmetic of function fields could lead, via construction of new error-correcting and cryptographic codes, and expanders to the design of faster algorithms in computer science used in everyday applications. To achieve this aim the mathematicians involved will communicate their results to computer scientists. This will be achieved in the way described above: publications in journals, conference talks and personal communication with leading computer scientists.