Splicing Summation Formulae and Triple Product L-Functions

该奖项涉及朗兰兹计划，该计划被描述为数学中的大统一理论。在某种意义上，该理论的原子是自同构的表示。朗兰兹功能猜想预言，自然对应关系的集合保存了这些原子。为了准确地表述这一猜想，需要各种数学学科，如数论、表示论、调和分析、代数几何和数学物理。反过来，对猜想的研究丰富了这些主题，在某些情况下还完全重塑了它们。保持自同构表示的对应的一个特别重要的例子是自同构张量积。众所周知，为了建立朗兰兹函数的这种特殊情况，只需证明某些被称为L函数的函数在分析上是良好的就足够了。最近，Braverman和Kazhdan，Ngo，Lforguge和Sakellaridis解释说，如果能得到某些广义Poisson求和公式，这些L函数的预期性质就会遵循。PI分离了一个特定的已知泊松求和公式家族，并建议将它们拼接在一起，以获得与建立自同构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优势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.