Analytic Number Theory at the Interface

该奖项的主要目的是发展解析数论与其他数学领域之间的联系，特别是动力学和概率论。动力学中的一个基本问题是理解一个确定性过程的随机化速度有多快，例如，以规则间隔采样的气体中的粒子。规则间隔相当于在整数处对系统进行采样。要回答这个领域中比较微妙的问题，通常需要对整数的性质有深刻的理解。相反，数论中的一个基本问题是证明整数的基本性质（例如它们的质因数分解）是随机的，在这里，来自动力学的技术可能是有用的。至于概率，来自数学物理的问题导致了相互作用系统研究技术的发展，例如约束在表面上的电子气体的宏观性质。最近人们了解到，这些技术在数论的背景下是适用的，例如在黎曼ζ函数的研究中，黎曼ζ函数是控制整数精细性质的基本函数。本建议的第二个目标是在数论的背景下进一步发展这些技术。PI将继续在与本研究相关的主题上培训研究生和指导博士后。在动力学方面，这个项目的重点是理解动力系统在空间中每一点的稀疏集（素数，平方）上的收敛性，特别关注环流。第二个目标是从圆方法的角度重新审视Elkies-McMullen关于以1为模的整数的平方根的间隙分布的工作。在概率论中，重点是发展分支随机漫步和统计力学之间的进一步联系，特别是通过关注Fyodorov-Hiary-Keating猜想及其其他特征。该项目的最终目标是发展我们对分数权重的自同构形式的理解，并将其应用于具体的数论问题（例如Kummer和的均匀分布）。该奖项反映了美国国家科学基金会的法定使命，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。

The main objective of this award is to develop connections between analytic number theory and other areas of mathematics, specifically dynamics and probability theory. A basic problem in dynamics is to understand how quickly a deterministic process, for example particles in a gas sampled at regularly spaced intervals, randomizes. The regular spacing amounts to sampling the system at the integers. Answering the subtler questions in this area often requires a non-trivial understanding of the properties of the integers. Conversely, a basic problem in number theory is to show that fundamental properties of the integers (e.g their prime factorization) randomize, and here techniques from dynamics can be useful. As for probability, questions from mathematical physics led to the development of techniques for the study of interacting systems, for example the macroscopic properties of a gas of electrons constrained to a surface. It has been recently understood that these techniques are applicable in a number theoretic context, for example in the study of the Riemann zeta-function, a basic function governing the finer properties of the integers. A second objective of this proposal is to develop such techniques further in a number theoretic context. The PI will continue training graduate students and mentor postdocs on topics related to this research.In dynamics the focus of this project is to understand the convergence of dynamical systems over sparse sets (primes, squares) at every point in the space, with a particular focus on horocycle flow. A second goal is to reexamine the work of Elkies-McMullen on the gap distribution of square-roots of integers modulo one from the perspective of the circle method. In probability theory, the focus is on developing further connections with branching random walks and statistical mechanics, specifically by focusing on the Fyodorov-Hiary-Keating conjecture and its other avatars. A final goal of the project is to develop our understanding of automorphic forms of fractional weight, with applications to concrete number theoretic problems (e.g the equidistribution of Kummer sums).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.