Workshop: Towards a Local Proof of the Local Langlands Correspondence

University Of Chicago, Chicago IL

Investigators

Abstract

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.