Analysis of Discrete Structures and Applications

这个项目处理的离散结构的分析范围广泛的问题，从经典的问题，在解析数论有关差距分布之间的素数和数域的形状，混合行为的随机游动组。 一个中心主题是从解析数论，概率和组合学的技术之间的相互作用。 首席研究员最近取得了突破，利用概率技术解决了Paul Erdos在组合数论中的一个著名问题，部分调查将开发这一结果的应用。 所研究的问题在密码学和量子计算中有应用。 资助将用于支持一组本科生和研究生的研究，并支持针对对解决问题感兴趣并开始研究的高中生和本科生的推广工作。该项目包括自守形式，筛选方法和群体随机游动分析的问题。 这些项目分为纯解析数论的主题，例如使用自守形式来研究数域整数环的晶格形状，使用解析数论中常见技术的项目，如指数和和振荡积分理论，研究马尔可夫链的渐近混合和其他概率极限问题，以及使用更纯粹概率技术的项目。 该项目的一个重要组成部分扩展了理论的zeta函数的预齐次向量空间在研究低度数领域。 各种现代分析技术将被采用，包括使用自守形式的高阶群体，分区函数从统计力学，点的分布在nilmanifold，和理论的产品随机matrix.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

This project treats a broad range of problems in the analysis of discrete structures ranging from classical problems in analytic number theory concerning the gap distribution between primes and the shape of number fields, to the mixing behavior of random walks on groups. A central theme is the interaction between techniques from analytic number theory, probability, and combinatorics. The principal investigator recently made a breakthrough by using probabilistic techniques to solve a famous problem of Paul Erdos in combinatorial number theory, and part of the investigation will develop applications of this result. The problems studied have applications to cryptography and quantum computation. Funding is to be used to support the research of a group of undergraduate and graduate students and to support outreach efforts aimed at high school and undergraduate students interested in problem solving and beginning research.The project includes problems on automorphic forms, sieve methods, and the analysis of random walks on groups. The projects are split between topics in pure analytic number theory, such as the use of automorphic forms to study the lattice shape of the ring of integers of number fields, projects which use techniques common in analytic number theory, like the theory of exponential sums and oscillatory integrals, to study the asymptotic mixing of Markov chains and other limit problems in probability, and projects which use more purely probabilistic techniques. A significant part of the project extends the theory of zeta functions of prehomogeneous vector spaces in studying low degree number fields. A variety of modern analytic techniques are to be employed, including the use of automorphic forms on higher rank groups, partition functions from statistical mechanics, the distribution of points on nilmanifolds, and the theory of products of random matrices.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.