Summation Formulae and Triple Product L-functions in Higher Rank

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The research supported under this award is aimed at fundamentally advancing our understanding of the Langlands functoriality conjecture. The Langlands functoriality conjecture is a profound unifying force in modern mathematics. Though largely open, even what is currently known has consequences that ripple throughout number theory, representation theory, algebraic geometry, and even mathematical physics. The basic objects of study are automorphic representations. Automorphic representations are defined using locally symmetric spaces, which are highly symmetric spaces that can be thought of as arbitrary dimensional drums. Every sound on a drum is a superposition of pure tones and automorphic representations are analogous to these pure tones. These are very complicated objects. Every locally symmetric space has a comparatively simple invariant called an L-group, and the functoriality conjecture predicts that when one can relate L-groups one can relate (the far more complicated) automorphic representations. The PI will also train graduate students in this area of research and is involved in mentoring activities at all levels from high school through to post-doctoral.Using relations between L-groups one defines functions called L-functions, and proving L-functions are "nice" is enough to prove Langlands functoriality. This proejct will develop new techniques to prove that L-functions are nice. The oldest tool available for proving L-functions are nice is the Poisson summation formula familiar from Fourier analysis, signal processing, etc. The PI and B. Liu have recently discovered generalizations of the Poisson summation formula that are the first of their kind. The more particular goal of the research is to extend and exploit these generalizations with the aim of studying triple product L-functions in higher rank. Using converse theory as developed by Cogdell and Piatetski-Shapiro many cases of Langlands functoriality can be reduced to the study of triple product L-functions so this is of fundamental importance.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.