CAREER: Building bridges between number theory and harmonic analysis

职业：在数论和调和分析之间架起桥梁

• 批准号: 2237937
• 负责人: Theresa Anderson
• 金额: $ 53.14万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2028-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2237937&HistoricalAwards=false
• 关键词: CAREER; Building; bridges; between; number

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.

数学的两个领域通常被视为独立的是数论和分析。数论处理整数的性质，例如快速将大数分解为素数，并支持计算机安全和黑洞的行为。谐波分析可以被看作是数学上用未成形的粘土制作雕塑;这个领域需要一个复杂的函数，并将其分解成简单的部分-它的谐波-并且是医学成像和量子态的核心。想象和建立一个新的景观，有时被认为是断开的区域可以协调是这个项目的最前沿：而不是利用已知的连接，新的连接将围绕三个不同的区域创建。首先，将进行曲率离散算子的定量行为;虽然与许多应用相关，但这里的新奇将是从两个领域的角度开发使用分析和数论的基础设施。第二部分将继续发现隐藏的数论性质的框架下谐波分析，进一步这两个领域的数学网格系统。最后，确定一个随机的代数对象的行为，如多项式，是数学中的一个中心问题。通过将傅立叶分析集成到这个领域，将能够以各种各样的方式更好地预测这种随机性。本科生将大量参与研究，特别是在第二部分，通过夏季和学期的研究和培训计划。多方面和网络建设的教育计划还包括一个名为“拓宽边缘：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，公平和包容办公室合作开发。该项目包括三个主要项目，都是关于调和分析和数论之间的交叉。首先，得到了曲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，而离散环境则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，这些边界连接到格点计数和信息在这方面将量化通过精细分析的指数和，包括离散的限制问题的各种表面的平移膨胀不变。其次，继续揭示隐藏的数论结构和技术领域的二元分析将追求，导致无数的结构定理。第三，算术统计中的傅立叶分析技术的程序将使用来自分析的新见解来改进各种代数兴趣的计数，例如字段和多项式。先前的调查已经成功，最近的结果包括一个重大步骤，解决猜想的货车德尔Waerden的随机多项式和计数的有限判别伽罗瓦群。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

A framework for discrete bilinear spherical averages and applications to $\ell ^p$-improving estimates

离散双线性球面平均值的框架及其在 $ell ^p$ 改进估计中的应用

• DOI: 10.4064/cm9216-1-2024
• 发表时间: 2024
• 期刊: Colloquium Mathematicum
• 影响因子: 0.4
• 作者: Anderson, Theresa C.;Kumchev, Angel V.;Palsson, Eyvindur A.
• 通讯作者: Palsson, Eyvindur A.