# Motivic Integration, Orbital Integrals, and Zeta-Functions (14w5155)

Arriving Sunday, November 30 and departing Friday December 5, 2014

## Organizers

William Casselman (University of British Columbia)
Julia Gordon (University of British Columbia)
Francois Loeser (Institut Mathematique de Juisseu)

## Objectives

The proposed workshop is intended to bring together two distinct mathematical groups -- those working in harmonic analysis on $p$-adic groups, and those in motivic integraction. The overall objective of the workshop is to understand $p$-adic orbital integrals better.

The first specific goal is to enable the members of one group to understand the techniques of the other. The second goal is to make the two communities aware of each other's open problems, especially the problems amenable to motivic integration techniques. We expect that some of the problems would in fact be solved during the conference.

The third, less precise goal, is to explore the connection between orbital integrals, arc spaces, and (motivic) Igusa zeta-functions.

The final goal is to produce concrete output in the form of lecture notes. We shall start with a series of introductory talks on (a) motivic integrals, (b) Igusa zeta-functions, (c) arc spaces, vanishing cycles, and Bernstein polynomials. After that, we intend that all the other participants give talks. We shall ask the speakers to produce lecture notes for these introductory talks, thus filling a large gap in the literature. There are several recent Ph.D.s on the list of invited participnts (please see below); we hope that the workshop will help with the development of their research programmes.

Motivic integration is a relatively new approach, which has developed a lot in the last two years. In view of the recent breakthroughs in the study of the orbital integrals in the function field case, now seems to be a good time to bring the two communities working on these problems together.

section*{Public announcement} This workshop will explore the connections between two areas of mathematics which until 10 years ago appeared completely unrelated. The first area is model theory -- a modern branch of mathematical logic, which lead to the transfer principle by Ax and Kochen in the 70s. The second area is harmonic analysis on $p$-adic groups, a field that plays a major role in number theory, for example, in the proof of Fermat's last theorem. In recent years it has turned out that some methods of model theory, by expanding our notions of integration, apply to the problems in harmonic analysis on $p$-adic groups.

begin{thebibliography}{99}

bibitem{arthur:gln} {Arthur, James.} {The endoscopic classification of representations: orthogonal and symplectic groups}. {sl to be published by AMS}, {(2012)}.

bibitem{assem:rationality} {Assem, Magdy}. {A note on rationality of orbital integrals on a $p$-adic group}. {sl Manuscripta Math.}, {89}, {267-279}, {1996}.

bibitem{cluckers-loeser} {Cluckers, Raf} and {Loeser, Fran{c{c}}ois}. {Constructible motivic functions and motivic integration}. {sl Invent. Math.}, {173},{(1)}, {23--121}, {2008}.

bibitem {cluckers-loeser:fourier} {Cluckers, Raf} and {Loeser, Fran{c{c}}ois}. {Constructible exponential functions, motivic Fourier transform and transfer principle}. {sl Annals of Mathematics}, {171 (2)}, {2010}, {1011--1065}.

bibitem{cluckers-hales-loeser} {Cluckers, Raf}, {Hales, Thomas}, and {Loeser, Fran{c{c}}ois}. {Transfer principle for the {F}undamental {L}emma}. {sl On the Stabilization of the Trace Formula, edited by L. Clozel, M. Harris, J.-P. Labesse, B.-C. Ng^o}, {International Press of Boston}, {2011}.

bibitem{CGH-1} {Cluckers, Raf}, {Gordon, Julia}, and {Halupczok, Immanuel}. {Integrability of oscillatory functions on local fields: transfer principles}. arXiv:1111.4405, {2011}.

bibitem{CGH-2} {Cluckers, Raf}, {Gordon, Julia}, and {Halupczok, Immanuel}. {Local integrability results in harmonic analysis on reductive groups in large positive characteristic}. arXiv:1111.7057, {2011}.

bibitem{denef-loeser:igusa} {Denef, Jan} and {Loeser, Fran{c{c}}ois}. {Motivic Igusa zeta-functions}. {sl J. Algebraic Geom.}, {7}, {1998}, {505-537}.

bibitem{HK} {Hrushovski, E. and Kazhdan, D.}. {Integration in valued fields}. {Algebraic geometry and number theory}. {sl Progr. Math.}, {253}, {261--405}, {Birkh"auser Boston}, {Boston, MA}, {2006}.

bibitem{loeser:bernstein} {Loeser, Fran{c{c}}ois}. {Fonctions D'Igusa p-adiques et Polynomes de Berstein}. {sl American Journal of Mathematics}. {110 (1)}, {1988}, {1-21}.

bibitem{ST} {Sug Woo Shin} and {Nicolas Templier}. {Sato-Tate theorem for families and low-lying zeroes of $L$-functions}. With appendices by R. Kottwitz and R. Cluckers, J. Gordon and I. Halupczok. http://arxiv.org/abs/1208.1945. bibitem{yun:FLJR} {Yun, Zhiwei}. {Fundamental Lemma of Jacquet-Rallis}. With appendix by Julia Gordon. {sl Duke Math J.}, {156 (2)}, {2011}, {220-227}.

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