Analytic Number Theory at the Interface

该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了解确定性过程（例如以规则间隔采样的气体中的粒子）随机化的速度。规则的间隔相当于以整数对系统进行采样。在这个领域中，要解决更微妙的问题，通常需要对整数的性质有一个不平凡的理解。相反，数论中的一个基本问题是证明整数的基本性质（例如它们的素因子分解）是随机的，这里来自动力学的技术可能是有用的。至于概率，数学物理学的问题导致了研究相互作用系统的技术的发展，例如电子被约束到表面的气体的宏观性质。最近人们已经认识到，这些技术适用于数论背景下，例如在黎曼zeta函数的研究中，这是一个控制整数的更精细性质的基本函数。本建议的第二个目标是在数论背景下进一步发展这种技术。PI将继续培训研究生和导师博士后有关本研究的主题。在动力学中，该项目的重点是了解稀疏集（素数，平方）在空间中的每一点上的动力系统的收敛性，特别关注horocycle流。第二个目标是重新审查工作的埃尔金斯-麦克马伦的差距分布的平方根整数模1的角度来看，圆的方法。在概率论中，重点是发展与分支随机游动和统计力学的进一步联系，特别是通过关注Fyodorov-Hiary-Keating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自守形式的理解，并将其应用于具体的数论问题（例如库默和的均匀分布）。该奖项反映了NSF的法定使命，并被认为值得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来支持。

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.