Analytic Number Theory over Function Fields

数论是一个数学领域，它以普通的计数及其在相加和相乘时的行为为中心。虽然这一领域的问题往往很容易陈述，但它们可能非常难以解决。函数域数论的子域旨在通过考虑一种模型或平行宇宙来洞察这些问题，在这种模型或平行宇宙中，数字的行为不同。我们认为将数字相加或相乘时发生的事情是正常的，但只是简单地去掉了多余的部分，而不是进位。当然，有了这个修改后的规则，算术变得更容易了，但更令人惊讶的是，数论中一些最重要的问题也变得更容易了，甚至一些最难的问题也变得可以解决了。(从技术上讲，我们应该使用二进制或任何质数基数，而不是通常的基数10。)或者，我们可以将这种变型算术描述为多项式函数在单变量中的加法或乘法。在这种情况下，我们可以通过将多项式的图形视为几何对象来将数论问题与几何联系起来。在这个奖项中，PI的研究使用几何工具来解决这一领域的新问题。PI的研究解决了数论中经典问题的函数场类似问题，包括孪生素数猜想和Chowla猜想(都与Shusterman联合)，Ramanujan猜想的情况(与Templier联合)，以及关于L函数的矩的猜想。在这个奖项中，PI将沿着这些路线继续下去，证明关于素数分布、L函数矩和自同构形的更多结果，并在更深的方向上工作，如非阿贝尔科恩-列斯特拉启发式。这些工作都是基于上同调理论，其中的基本结果，Deligne的Riemann假设，允许许多不同的分析问题(关于证明某些不等式的问题)归结为上同调问题(关于计算一个簇或簇的一些上同调群的问题)。相关的变种是高维的，计算必要的上同调群需要像消失循环理论和特征循环这样的技术。这个奖项反映了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。

Number theory is an area of mathematics that centers on the ordinary counting numbers and their behavior when we add and multiply them. While problems in this area are often simple to state, they can be fiendishly difficult to solve. The subfield of function field number theory aims to obtain insight on these problems by considering a kind of model or parallel universe where numbers behave differently. We consider what happens when we add or multiply numbers as normal but, except, instead of carrying digits, we simply drop the excess. Certainly arithmetic is a little easier with this modified rule, but more surprisingly, some of the most important problems in number theory become easier as well, with even some of the most difficult ones becoming solvable. (Technically, we should work in binary, or any prime base, rather than our usual base 10, for this.) Alternately, we can describe this variant arithmetic as the addition or multiplication of polynomial functions in a single variable. In this setting, we can connect number-theoretic questions to geometry, by viewing the graph of the polynomial as a geometric object. In this award the PI's research uses geometric tools to solve new problems in this area.The PI's research has resolved function field analogues of classical problems in number theory, including the twin primes conjecture and Chowla's conjecture (both joint with Shusterman), cases of the Ramanujan conjecture (joint with Templier), and conjectures about moments of L-functions. In this award the PI will continue along these lines, proving additional results about the distribution of prime numbers, L-function moments, and automorphic forms, and work in further directions such as non-abelian Cohen-Lenstra heuristics. These works are all based on etale cohomology theory, where the foundational result, Deligne's Riemann Hypothesis, allows many different analytic problems (problems about proving some inequality) to be reduced to cohomology problems (problems about calculating some of the cohomology groups of a variety or sheaf). The relevant varieties are high-dimensional, and calculating the necessary cohomology groups requires techniques like vanishing cycles theory and the characteristic cycle.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.