Analytic Number Theory over Function Fields

数论是数学的一个领域，它集中在普通的计数数字和他们的行为，当我们添加和乘以他们。虽然这方面的问题通常很容易陈述，但它们可能非常难以解决。函数域数论的子领域旨在通过考虑一种模型或平行宇宙来洞察这些问题，其中数字表现不同。我们把数字的加法或乘法看作是正常的，但是，除了不携带数字之外，我们只是简单地丢弃多余的数字。当然，用这个修改过的规则，算术会容易一些，但更令人惊讶的是，数论中一些最重要的问题也变得容易了，甚至一些最困难的问题也变得可以解决了。（从技术上讲，我们应该在二进制或任何素数基中工作，而不是我们通常的基数10。或者，我们可以将这种变体算术描述为单个变量中多项式函数的加法或乘法。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，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解决了这一领域的新问题。PI的研究解决了数论中经典问题的函数场类似物，包括孪生素数猜想和Chowla猜想（均与Shusterman联合），Ramanujan猜想的情况（与Templier联合），以及关于L函数矩的图解。在这个奖项中，PI将继续沿着这些路线，证明有关素数分布，L-函数矩和自守形式的其他结果，并在进一步的方向工作，如非阿贝尔科恩-伦斯特拉几何学。这些工作都是基于etale上同调理论，其中的基本结果，德利涅的黎曼假设，允许许多不同的分析问题（问题证明一些不等式），以减少到上同调问题（问题计算的一些上同调群的品种或层）。该奖项反映了NSF的法定使命，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。

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.