Arithmetic Combinatorics and Applications

该项目的研究领域是算术组合学。随着解析数论、傅立叶分析、概率、计算机科学(如椭圆曲线密码学和伪随机数发生器)，甚至理论物理(如固体物理)的许多新的应用，这已成为一个跨学科的研究领域。虽然算术组合学中的某些主题在数论中是经典的，但也有一些新的结构问题被证明是重要的。一些有用的工具(例如指数和和估计和特征和估计)本身对研究人员很有吸引力。有限域上的组合问题仍然提供了许多挑战，特别是涉及有限域上簇上的点的阶的问题(例如，与Markoff曲面有关的最新发展)。这项研究涉及不同的人群和不同数学分支的互动，有可能在一些众所周知的悬而未决的问题上取得进展。首席调查员将通过解释组合数论的研究，每周举办一次研究生级别的研讨会，讨论分析和组合学的基本方法。这个项目的目标是继续研究节点交的普适性，有限域的乘法群中的算术级数，非线性Roth型定理，点的阶和密度，有限域上的簇，以及不完全和短特征和理论。除了(离散的)傅立叶分析和概率中的常用工具外，还使用了算术组合学的技术，特别是在广义算术级数和定量Nullstellensatz中的各种形式的因式分解、指数和和的和积理论以及特征和估计(包括混合特征和、非常短间隔上的特征和以及这些具有不同模或变元的和)。事实证明，在各种环境下求和乘积的结果本身是有意义的，它们导致了解析数理论的新结果和提出的一些问题。这一奖项反映了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 research area of the project is arithmetic combinatorics. This has become an interdisciplinary field of research with many new emerging applications to analytic number theory, Fourier analysis, probability, computer science (e.g. elliptic curve cryptography, and pseudorandom number generators), and even theoretical physics (e.g. solid state physics). While certain themes in arithmetic combinatorics are classical in number theory, there is also focus on new structural questions that turned out to be important. Some of the useful tools (e.g. exponential sum and character sum estimates) are themselves attractive to researchers. Combinatorial problems in finite fields continue to offer many challenges, in particular questions involving orders of points on varieties over finite fields (e.g. recent developments related to the Markoff surface). This research involves different groups of people and the interaction of various branches of mathematics, with the potential of making progress on some well-known unsolved problems. The principal investigator will run a weekly seminar at graduate level on basic methods in analysis and combinatorics through explanation of research in combinatorial number theory. Special efforts would be made to attract first-generation college students.The goal of this project is to continue research on universality for nodal intersections, arithmetic progressions in multiplicative groups of finite fields, non-linear Roth type theorem, orders and density of points, on varieties over finite fields, and the theory of incomplete and short character sums. Besides the usual tools in (discrete) Fourier analysis and probability, techniques from arithmetic combinatorics were used, particularly, various versions of factorization in generalized arithmetic progressions and quantitative Nullstellensatz, sum-product theory for the exponential sum and character sum estimates, (including mixed character sums, character sums over very short intervals, and these sums with various moduli or arguments). It turns out that results from sum-product in various settings are of interest in their own right as well as they lead to new results in analytic number theory and some of the problems proposed.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.