The Subconvexity Problem

素数在数学中具有根本的重要性，因为每个数都可以被唯一地写成素数的乘积。素数的一个神秘特征是，它们如何分布的统计数据与随机数字序列产生的结果相似。尽管一个数是否是质数在任何方面都不是随机的。L函数是数学对象，它在理论上允许我们理解素数和其他数论对象的统计特征。这个项目研究了关于L-函数的一个重要猜想，称为次凸性问题，它指出这些L-函数所取的值小于预期。如果这是真的，这将是一个重要的一步解锁的统计信息，L-函数包含。这个奖项的目的是取得进展的次凸性问题的L-函数在更高的排名，并证明次凸性的范围尽可能广泛的自守形式。该项目将侧重于U（n+1）x U（n）和GL（n+1）x GL（n）组。PI将使用算术放大、表征理论和来自微局部分析的思想来解决问题。微型地方电梯将发挥关键作用。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Prime numbers are of fundamental importance in mathematics, as every number can be uniquely written as a product of primes. A mysterious feature of prime numbers is that the statistics of how they are distributed is similar to what a random sequence of numbers would produce. This is despite the fact that whether a number is prime or not is in no respect random. L-functions are mathematical objects that conjecturally allow us to understand this statistical feature of primes and other objects from the theory of numbers. This project investigates an important conjecture about L-functions, known as the subconvexity problem, which states that the values taken by these L-functions are smaller than expected. If true, it would be an important step in unlocking the statistical information that L- functions contain.The aim of this award is to make progress on the subconvexity problem for L-functions in higher rank, and to prove subconvexity for as wide a range of automorphic forms as possible. The project will focus on the groups U(n+1) x U(n) and GL(n+1) x GL(n). The PI will approach the problem using arithmetic amplification, representation theory, and ideas from microlocal analysis. A key role will be played by microlocal lifts. The methods used will not depend on the rank of the group in an essential way.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.