Questions at the Interface of Analysis and Number Theory

分析与数论的交叉问题

• 批准号: 2231990
• 负责人: Theresa Anderson
• 金额: $ 18万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2231990&HistoricalAwards=false
• 关键词: Questions; Interface; Analysis; Number; Theory

Harmonic analysis and number theory are fundamental fields of mathematics that are used to describe and interpret many real-world phenomena. Harmonic analysis involves breaking up a mathematical object such as a function into pieces that are easier to understand. The beauty of this area is that the pieces are oftentimes simple, yet represent the whole with accuracy. Number theory involves deceptively simple statements about the integers, easy to test, yet often difficult to prove. Though seemingly disparate, analysis and number theory share many interactions. For instance, one can use intricate analysis of complex functions to answer fundamental questions about prime numbers. This project explores a variety of problems at the interface of these two areas. In particular, the PI will consider discrete variants of operators in analysis, which enjoy applications in fields such as medical imaging and cosmology. To analyze these operators, continuous techniques often fail, and one has to develop number theoretic techniques adapted to the underlying geometry of the analytic problem. The PI seeks to provide new bounds, new techniques, sharper analysis and broader connections. The PI also plans to bring Fourier analysis, a fundamental decomposition of the time-frequency domain, such as that used to understand waves, into the emerging field of arithmetic statistics. Here she seeks to provide sharp counts of a wide variety of objects of arithmetic interest, such as elliptic curves used in cryptography. As a broader impact, the PI will spark new mathematical conversations between analysts and number theorists and also improve the educational and scientific climate for underrepresented groups.This project addresses several fundamental questions at the interface of analysis and number theory. Firstly, the PI pursues bounds for discrete variants of continuous operators in harmonic analysis that involve integration over a curved subvariety. These bounds provide quantitative distributional facts about the underlying Diophantine equations that define these varieties, which makes them different from their continuous counterparts. In particular, since continuous techniques usually do not carry over in this setting, the PI will develop refined number theoretic techniques to bound several operators, including multilinear spherical variants, variants defined over the primes, and higher codimensional analogues. In particular, the higher codimensional study should open new avenues of problems as very little is known in this setting. Solving these problems has connections to discrete geometry, lattice point counts of surfaces, and Falconer's distance conjecture. In another series of problems, the PI will pursue "sparse bounds" for both continuous and discrete operators. Sparse bounds are a refinement of Lebesgue space bounds that allow one to deduce weighted estimates. Finally, the PI plans to pursue a far reaching program in arithmetic statistics. This is an area greatly developed on the algebraic side recently. The PI plans to inject Fourier analytic techniques to obtain precise lattice point counts that are adaptable to take advantage of the power of the algebraic techniques and push those bounds even further. In particular, the PI hopes to obtain counts on certain objects such as elliptic curves, with an eye to not only developing techniques, but also fostering interactions between number theorists and analysts in new ways.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.

调和分析和数论是数学的基础领域，用于描述和解释许多现实世界的现象。 调和分析涉及将数学对象（例如函数）分解成更容易理解的部分。 这个区域的美妙之处在于，这些部件通常很简单，但却准确地代表了整体。 数论涉及关于整数的看似简单的陈述，易于测试，但通常难以证明。 尽管分析和数论看似不同，但它们有许多相互作用。例如，人们可以使用复杂函数的复杂分析来回答有关素数的基本问题。 该项目探讨了这两个领域交界处的各种问题。 特别是，PI 将在分析中考虑算子的离散变体，这些变体在医学成像和宇宙学等领域有着广泛的应用。 为了分析这些算子，连续技术通常会失败，并且必须开发适合分析问题的基础几何的数论技术。 PI 寻求提供新的界限、新技术、更敏锐的分析和更广泛的联系。 PI 还计划将傅里叶分析（一种时频域的基本分解，例如用于理解波的分析）引入新兴的算术统计领域。 在这里，她试图提供各种算术感兴趣的对象的精确计数，例如密码学中使用的椭圆曲线。 作为更广泛的影响，PI 将引发分析师和数论学家之间新的数学对话，并改善代表性不足群体的教育和科学氛围。该项目解决了分析和数论界面的几个基本问​​题。 首先，PI 在调和分析中追求连续算子的离散变体的界限，其中涉及对曲线子变体的积分。 这些界限提供了有关定义这些变体的基础丢番图方程的定量分布事实，这使得它们不同于连续的对应物。 特别是，由于连续技术通常不会在这种情况下继续使用，因此 PI 将开发精细的数论技术来限制多个算子，包括多线性球面变体、在素数上定义的变体以及更高的共维类似物。 特别是，更高维度的研究应该开辟解决问题的新途径，因为在这种情况下我们知之甚少。解决这些问题与离散几何、表面格点计数和福尔科纳距离猜想有关。 在另一系列问题中，PI 将为连续和离散算子追求“稀疏边界”。 稀疏界限是勒贝格空间界限的一种改进，允许人们推导出加权估计。 最后，PI 计划开展一项影响深远的算术统计项目。 这是最近在代数方面得到极大发展的一个领域。 PI 计划注入傅里叶分析技术来获得精确的格点计数，这些点计数可以利用代数技术的力量并进一步突破这些界限。 特别是，PI 希望获得椭圆曲线等某些物体的计数，不仅着眼于开发技术，而且还以新的方式促进数论学家和分析师之间的互动。该奖项反映了 NSF 的法定使命，并被认为是值得的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来获得支持。

项目成果

Discrete Maximal Operators Over Surfaces of Higher Codimension

高维表面上的离散最大算子

• DOI: 10.1007/s44007-021-00017-4
• 发表时间: 2022-06
• 期刊: La Matematica
• 作者: Anderson, Theresa C.;Kumchev, Angel V.;Palsson, Eyvindur A.
• 通讯作者: Palsson, Eyvindur A.

Quantitative Hilbert Irreducibility and Almost Prime Values of Polynomial Discriminants

多项式判别式的定量希尔伯特不可约性和几乎素值

• DOI: 10.1093/imrn/rnab296
• 发表时间: 2021-11
• 期刊: International Mathematics Research Notices
• 影响因子: 1
• 作者: Anderson, Theresa C;Gafni, Ayla;Lemke Oliver, Robert J;Lowry;Shakan, George;Zhang, Ruixiang
• 通讯作者: Zhang, Ruixiang

Bounds on 10th moments of (x, x^3) for ellipsephic sets

椭圆集 (x, x^3) 的 10 阶矩的界限

• 发表时间: 2024-03
• 期刊: Contemporary mathematics American Mathematical Society
• 作者: Anderson, Theresa;Hu, Bingyang;Liu, Yu;Talmage, Alan
• 通讯作者: Talmage, Alan

On the translates of general dyadic systems on $${{\mathbb {R}}}$$

关于 $${{mathbb {R}}}$$ 上一般二元系统的翻译

• DOI: 10.1007/s00208-019-01951-z
• 发表时间: 2020-08
• 期刊: Mathematische Annalen
• 影响因子: 1.4
• 作者: Anderson, Theresa C.;Hu, Bingyang;Jiang, Liwei;Olson, Connor;Wei, Zeyu
• 通讯作者: Wei, Zeyu

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: 2024-01
• 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.