Analytic Number Theory at the Interface

这个奖项的主要目的是发展解析数论和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确定性过程的随机化速度，例如以规则间隔采样的气体中的粒子。规则间隔相当于以整数对系统进行采样。回答这一领域的更微妙的问题通常需要对整数的属性有很好的理解。相反，数论中的一个基本问题是证明整数的基本性质(例如，它们的素因式分解)是随机化的，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研究相互作用系统的技术的发展，例如约束在表面上的电子气的宏观性质。最近已经理解，这些技术可应用于数论环境中，例如在Riemann Zeta函数的研究中，Riemann Zeta函数是控制整数的更精细性质的基本函数。这项提议的第二个目标是在数论的背景下进一步发展这种技术。PI将继续就与此研究相关的主题培训研究生和博士后导师。在动力学中，本项目的重点是了解稀疏集(素数、平方)上的动力系统在空间中的每一点的收敛，特别关注周期流。第二个目标是从圆法的角度重新审视Elkies-McMullen关于模1整数平方根间隙分布的工作。在概率论中，重点是进一步发展与分支随机游走和统计力学的联系，特别是通过关注费奥多罗夫-希里-基廷猜想及其其他化身。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NSF的法定使命，并通过使用基金会的智力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.