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.

数论和分析通常被视为独立的两个数学领域。数论研究整数的性质，例如将大数快速分解为素数，并支撑计算机安全和黑洞的行为。谐波分析可以被视为用数学方法用未成型的粘土制作雕塑；这个场将一个复杂的函数分解成简单的部分——它的谐波——并且是医学成像和量子态的核心。该项目的首要任务是想象和建造一个新的景观，使有时被认为互不相连的区域能够协调一致：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，将进行具有曲率的离散算子的定量行为；虽然与许多应用相关，但这里的新颖之处在于开发基础设施，以从两个领域的角度使用分析和数论。第二部分将继续发现调和分析框架中隐藏的数论特性，并在这两个领域进一步推进数学网格系统。最后，确定具有代数意义的随机对象（例如多项式）的行为是数学的中心问题。通过将傅立叶分析集成到这个竞争环境中，将能够以多种方式更好地预测这种随机性。本科生将通过夏季和学期的研究和培训计划大量参与研究，尤其是第二部分。这个多方面和网络建设的教育计划还包括教授一个名为“扩大利润：URM在数学科学中的经验”的研究生课程，该课程将与该机构的多样性、公平和包容性办公室合作开发。该项目由三个主要项目组成，全部涉及调和分析和数论的交叉点。首先，将获得涉及弯曲子流形积分的离散算子的算子界限。这是连续分析中的一个微妙话题，离散设置引入了新的挑战和障碍。数论工具的发展与分析相结合将极大地促进对这些对象的理解。此外，这些边界与格点计数相关，并且这方面的信息将通过指数和的精细分析来量化，包括平移膨胀不变的各种表面的离散限制问题。其次，将继续揭示二进分析领域中隐藏的数论结构和技术，从而产生无数的结构定理。第三，算术统计中傅立叶分析技术的程序将利用分析中的新颖见解来改进各种代数兴趣的计数，例如域和多项式。先前的研究已经取得了成功，因为最近的结果包括在解决范德瓦尔登关于随机多项式伽罗瓦群和有界判别式的数域计数方面的猜想方面迈出了重要一步。沿着这些思路的调查将继续进行。该奖项反映了 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。

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.