CAREER: Building bridges between number theory and harmonic analysis

通常被视为独立的两个数学领域是数论和分析。数论研究整数的性质，例如快速将大数分解成素数，并支持计算机安全和黑洞的行为。谐和分析可以被看作是用未成形的粘土数学地制作一个雕塑；这个场采用一个复杂的函数，并将其分解成简单的碎片--它的谐和--并且是医学成像和量子态的核心。想象和构建一种新的景观，有时被认为是互不相连的区域可以相互协调，这是该项目的首要任务：不是利用已知的联系，而是围绕三个不同的区域创建新的联系。首先，我们将介绍具有曲率的离散算子的量化行为；虽然与许多应用程序相关，但这里的新奇之处将是从这两个领域的角度来开发使用分析和数论的基础设施。第二部分将继续在调和分析的框架中发现隐藏的数论性质，在这两个领域进一步发展这个数学网格系统。最后，确定代数感兴趣的随机对象(如多项式)的行为是数学中的一个中心问题。通过将傅里叶分析整合到这一领域，将能够以各种方式更好地预测这种随机性。本科生将通过暑期和学期的研究和培训计划，深度参与研究，特别是在第二部分。这项多层面和网络建设的教育计划还包括一个名为“扩大差距：URM在数学科学中的经验”的研究生课程的教学，该课程将与该机构的多样性、公平性和包容性办公室合作开发。该项目包括三个主要项目，都是关于调和分析和数论的交叉。首先，得到了曲线子流形上涉及积分的离散算子的算子界。在连续分析中，这是一个微妙的话题，而离散的环境带来了新的挑战和障碍。与分析相结合的数论工具的发展将极大地推动对这些对象的理解。此外，这些边界与格点计数有关，这一前沿上的信息将通过对指数和的精细分析来量化，包括对各种平移膨胀不变的曲面的离散限制问题。其次，我们将继续探索并元分析领域中隐藏的数论结构和技术的发现，从而产生无数的结构定理。第三，算术统计中的傅立叶分析技术程序将使用分析中的新见解来改进各种代数兴趣的计数，如域和多项式。以前的研究已经取得了成功，因为最近的结果包括在解决van der Wairden关于Galois群的随机多项式的猜想和计数有界判别式的数域方面迈出了重要的一步。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Two areas of mathematics often viewed as separate are number theory and analysis. Number theory deals with properties of whole numbers, such as quickly factoring large numbers into primes, and underpins computer security and behavior of black holes. Harmonic analysis can be viewed as mathematically making a sculpture out of unformed clay; this field takes a complicated function and breaks it up into simple pieces – its harmonics – and is central to medical imaging and quantum states. Imagining and building a new landscape where areas sometimes perceived as disconnected can harmonize is at the forefront of this project: Instead of exploiting known connections, new ones will be created centered around three distinct areas. Firstly, quantitative behavior of discrete operators with curvature will be undertaken; while relevant to many applications, the novelty here will be developing the infrastructure to use both analysis and number theory from the perspective of both fields. The second part will be a continuation of the discovery of hidden number theoretic properties in the framework underlying harmonic analysis, furthering this mathematical grid system in both areas. Finally, determining the behavior of a random object of algebraic interest, such as a polynomial, is a central problem in mathematics. By integrating Fourier analysis into this playing field, will enable a better prediction of this randomness in a wide variety of ways. Undergraduates will be heavily involved in research, especially in the second part, via a summer and semester research and training program. The multifaceted and network-building education plan further includes the teaching of a graduate class intitled “Widening the margins: the experience of URMs in the mathematical sciences,” which will be developed in collaboration with the institution’s Diversity, Equity and Inclusion office.The project consists of three main programs, all on the intersection between harmonic analysis and number theory. Firstly, operator bounds for discrete operators involving integration over curved submanifolds will be obtained. This is a delicate topic in continuous analysis and the discrete setting introduces new challenges and obstacles. A development of number theoretic tools in tandem with the analysis will dramatically push forward understanding of these objects. Additionally, these bounds are connected to lattice point counts and information on this front will be quantified via refined analysis of exponential sums, including discrete restriction problems for a variety of surfaces that are translation dilation invariant. Secondly, a continued uncovering of a hidden number theoretic structure and techniques in the realm of dyadic analysis will be pursued, leading to a myriad of structure theorems. Thirdly, the program in Fourier analytic techniques in arithmetic statistics will use novel insights from analysis to improve a variety of counts of algebraic interest, such as fields and polynomials. Prior investigations here have already been successful as recent results include a major step in resolving a conjecture of van der Waerden on Galois groups of random polynomials and counting number fields of bounded discriminant. Investigations along these lines will be continued.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.