Randomness in number theory and automorphic forms

该提议涉及数论中的随机性。在研究丢番图问题时，随机性以多种形式出现，有时是透明的，负担是建立预期的随机性，而在其他情况下，发现随机性已经是一个主要问题。我们试图继续研究这一现象的具体环境包括与积分仿射群的轨道有关的丢番图问题，诸如莫比乌斯函数这样的序列中的随机性，特别是它与低复杂性动力系统的联系，自同构L函数族的零点随机性以及相应的自同构形式的零集(节点线)中的随机性。理解随机性问题是数论和算术量子混沌的许多应用的核心。数论和数学中的其他领域之间的相互作用有着悠久的传统。数学中的许多领域的发展部分是为了解决关于整数和素数的问题。自然，这些领域继续在数论中发挥核心作用。最新的是数学物理、理论计算机科学和信息理论之间的相互作用。这种交叉融合是这一提议的核心。例如，模曲面给出了经典混沌哈密顿量的独特例子，可以研究其量子化的细微之处。统计物理的技术，特别是随机矩阵系综模型，精确地描述了数论中被称为Zeta函数的中心对象家族的零点。组合学和理论计算机科学的思想，特别是“扩展图”，在理解一些经典的数论问题方面发挥了核心作用。

The proposal is concerned with randomness in number theory .Randomness appears in many forms when studying diophantine problems and it is sometimes transparent ,with the burden being to establish the expected randomness, while in other cases discovering the randomness is already a major issue . The specific settings in which we seek to continue to investigate this phenomenon include diophantine problems connected with orbits of groups of integral affine morphisms ,the randomness in the such sequences as the Mobius function and especially its connection to low complexity dynamical systems ,the randomness in zeros of families of automorphic L-functions and in the zero sets (nodal lines) of the corresponding automorphic forms. Understanding the randomness issue is at the heart of many applications to number theory and arithmetic quantum chaos .The interplay between number theory and other areas within mathematics has a long tradition.Many fields within mathematics were developed in part to solve problems concerning whole numbers and prime numbers. Naturally these areas continue to play a central role in the theory of numbers . More recent is the interplay between mathematical physics ,theoretical computer science and information theory .This cross fertilization is at the heart of this proposal .For example the modular surface gives perhaps the unique example of a classically chaotic hamiltonian for which the fine points of its quantization can be studied.Techniques from statistical physics and especially random matrix ensembles model precisely the zeros of families of central objects in number theory called "zeta functions" .Ideas from combinatorics and theoretical computer science ,notably "expander graphs", play a central role in understanding some classical number theoretical problems.