The organizers intend to hold a weekend workshop at the University of Illinois at Chicago, aimed at graduate students and young researchers, on the subject of the Local Langlands Correspondence. Recent work on the special fiber of the stable reduction of the Lubin-Tate tower by J. Weinstein using p-adic Hodge theory and p-divisible groups appears finally to make possible a purely local proof of the correspondence. Five experts will be invited to give talks on different pertinent subjects, which in total present the complete story behind the local correspondence: introductory material; C. Bushnell and P. Kutzko's type theory; M. Strauch's Jacquet-Langlands correspondence via the Lefschetz trace formula; Weinstein's aforementioned results on the stable reduction of the Lubin-Tate tower, and his joint work with M. Boyarchenko on the geometry of the special fiber. The workshop will be held on May 12-13, 2012. More information can be found at the conference website: http://math.uchicago.edu/~lxiao/workshop_site/
Number theory is the branch of pure mathematics devoted to the study of whole numbers and their relations; a particular goal is often to understand whether a given algebraic equation has a whole number solution. The apparent simplicity of such problems belies their complexity, and a box of intricate tools has been collected over the past 400 years to attack them. Chief among them is a 'correspondence' proposed by R. Langlands in the 1960s which, once established, would allow number theorists to exploit methods from a wide range of other, seemingly unrelated mathematical fields. For example, the 300 year old problem 'Fermat's Last Theorem' was successfully tackled by A. Wiles in the 1990s by establishing part of Langlands' correspondence. Now, although the correspondence remains mysterious in general, it can be split into pieces which can be studied 'one prime at a time', called 'local correspondences': since prime numbers are the atoms of the whole numbers, a common approach in number theory is to attack a problem one prime at a time. These local correspondences can be investigated using geometry and are much better understood; the main intent of the organizers is to host a weekend workshop where graduate students and young researchers can learn about the latest developments in the field.
The purpose of this NSF grant was to fund a weekend workshop in May 2012 at the University of Illinois at Chicago, aimed predominately at young mathematicians such as graduate students. The theme of this workshop was the so-called "Local Langlands Correspondence". This is a program of modern research in number theory which offers a bridge between algebraic techniques on one side, and geometric ideas on the other. Variations on this program have led to spectacular success in mathematical research in recent years; in particular, it was a fundamental tool in Andrew Wiles' celebrated solution of Fermat's Last Theorem in the 1990s. There have been particularly important developments in the field over the past ten years, offering hope that the present theory can be considerably simplified. Our aim in organizing this workshop was therefore to make young mathematicians aware of these developments, and to encourage them to conduct research in the field. This was achieved by inviting five experts, including some of those responsible for the most recent developments, to give a series of talks over the course of a weekend at the University of Illinois at Chicago. The workshop was attended by approximately 60-70 young mathematicians, 40 of whom were supported by this NSF grant. The participants came from a variety of mathematical fields, and we feel that the workshop encouraged interdisciplinary communication and germinated future collaborations, thus hopefully leading to new insights and breakthroughs which would not otherwise be possible.
