The goal of this proposal is to investigate the functoriality principle in the function field case. The starting points are Langlands' paper 'Beyond Endoscopy', Beilinson and Drinfeld's conjecture and the recent proof of the Fundamental Lemma to which the PI contributed in a crucial way. Strong ties among these elements will provide a new road to general functoriality in the function field case and will give direction to further investigation in the number field case as well.

Building on the PI's recent success in proving certain cases of the Fundamental Lemma, a central question in the Langlands program, the research program in this award offers a new set of deep and exciting problems for young researchers in the domain of algebraic geometry and representation theory.

Project Report

Publications. The P.I has coauthored three published articles on materials directly related to research supported by NSF Grant 1118716. (1) E. Frenkel; R. Langlands; B.C. Ngo: Formule des traces et fonctori- alit ?e: le d ?ebut d’un programme. Ann. Sci. Math. Qu ?ebec 34 (2010), no. 2, 199-243. (2) E. Frenkel; B. C. Ngo: Geometrization of trace formulas. Bull. Math. Sci. 1 (2011), no. 1, 129-199. (3) J. Heinloth; B. C. Ngo; Z. Yun: Kloosterman sheaves for reductive groups. Ann. of Math. (2) 177 (2013), no. 1, 241-310. The two papers in collaboration with Frenkel and Langlands spelled out a new strategy for proving the functoriality principle in the theory of automorphic representation as well as description of some explicit problems. In the paper in collaboration with Heinloth and Yun, we obtained a concrete application and solved a longstanding conjecture of N. Katz. The above mentioned works has been influential in setting a new direction for the field of automorphic representations, and espe- cially on the development of younger mathematicians working in this field: • Zhiwei Yun has found a spectacular solution to a conjecture of Serre in developing idea of the paper Heinloth-Ngo-Yun. • Ali Altug has pursued the approach proposed in Frenkel-Langlands- Ngo in a particular case and succeeded to establish the functoriality in that case. • Alexis Bouthier has developed ideas contained in Frenkel-Ngo to construct a new Hitchin fibration associated to the geometric side of the trace formula.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Andrew D. Pollington
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University of Chicago
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