Splicing Summation Formulae and Triple Product L-Functions

这个奖项涉及朗兰兹计划已被描述为一个大统一理论在数学。 在某种意义上，理论的原子是自守表示。 朗兰兹函性猜想预言，自然对应的集合保存了这些原子。 为了精确地表述这个猜想，需要涉及到数论、表示论、调和分析、代数几何和数学物理等多种数学学科。 反过来，关于猜想的工作丰富了这些主题，在某些情况下完全重塑了它们。保持自守表示的对应关系的一个特别重要的例子是自守张量积。一段时间以来，人们已经知道，为了建立朗兰兹函数性的这种特殊情况，只要证明某些被称为L-函数的函数是解析良好的就足够了。最近，布雷弗曼和Kazhdan，Ngo，Lafforgue和Sakellardan解释说，这些L-函数的预期性质将遵循，如果可以获得某些广义泊松求和公式。 PI已经分离出一个特定的已知泊松求和公式族，并建议将它们拼接在一起，以获得与建立自守张量积相关的泊松求和公式。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award concerns the Langlands program which has been described as a grand unification theory within mathematics. In some sense the atoms of the theory are automorphic representations. The Langlands functoriality conjecture predicts that a collection of natural correspondences preserve these atoms. To even formulate this conjecture precisely, mathematical subjects as diverse as number theory, representation theory, harmonic analysis, algebraic geometry, and mathematical physics are required. In turn, work on the conjecture has enriched these subjects, and in some cases completely reshaped them. One particularly important example of a correspondence that should preserve automorphic representations is the automorphic tensor product. It has been known for some time that in order to establish this particular case of Langlands functoriality it suffices to prove that certain functions known as L-functions are analytically well-behaved. More recently, Braverman and Kazhdan, Ngo, Lafforgue and Sakellaridis have explained that the expected properties of these L-functions would follow if one could obtain certain generalized Poisson summation formulae. The PI has isolated a particular family of known Poisson summation formulae and proposes to splice them together to obtain the Poisson summation formulae relevant for establishing the automorphic tensor product.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.