The main topic of the course is motivic integration, with a viewpoint which is close to p-adic integration, uniform in p. The goal of the series is to work towards the recent transfer principles for motivic integrals, which allow one to transfer several results from p-adic fields to local fields of positive characteristic and vice versa. We will develop the notions of motivic exponential functions, forming a natural class of functions which is stable under integration and under Fourier transformation. These functions, together with the transfer principles, have been recently used in applications to the Langlands program, for example (with Hales and Loeser) to derive the Fundamental Lemma in characteristic zero from the work of Ngo, and to show local integrability of Harish-Chandra characters in large positive characteristic, with Gordon and Halupczok. The subject combines techniques from algebraic/non-archimedean geometry, model theory, harmonic analysis and number theory.
After studying definable sets and functions in the Denef-Pas language in detail, we will define motivic (exponential) functions, and study p-adic integrals, uniformly in all p-adic fields. Then we will gradually build up towards proving the transfer principles of [Cluckers, Loeser: Constructible exponential functions, motivic Fourier transform and transfer principle, Annals of Mathematics, Vol. 171, No. 2, 1011-1065 (2010)] and of [Cluckers, Gordon, Halupczok: Integrability of oscillatory functions on local fields: transfer principles, to appear in Duke Math. J., arXiv:1111.4405]. A recent and introductory overview paper, including applications to the Langlands program, is [Cluckers, Gordon, Halupczok: Motivic functions, integrability, and uniform in p bounds for orbital integrals, to appear in Electronic Research Announcements in Math. (ERA), arXiv:1309.0594].
Time: Tuesday 10-12
Auditorium: HG G 43 (HWZ)
Begins: February 18, 2014