Questions at the Interface of Analysis and Number Theory

调和分析和数论是描述和解释许多现实世界现象的数学基础领域。调和分析涉及将一个数学对象(如函数)分解成更容易理解的部分。这个地区的美丽之处在于，这些碎片往往很简单，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，但往往难以证明。尽管分析和数论看似不同，但它们有许多共同之处。例如，人们可以使用复杂的复函数分析来回答关于质数的基本问题。这个项目探索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，这在医学成像和宇宙学等领域都有应用。为了分析这些算符，连续的技术常常失败，人们必须开发适应于分析问题的基本几何的数论技术。PI寻求提供新的界限、新的技术、更敏锐的分析和更广泛的联系。PI还计划将傅里叶分析引入新兴的算术统计领域。傅立叶分析是时频域的一种基本分解，例如用于理解波的分析。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。作为一个更广泛的影响，PI将在分析家和数论家之间引发新的数学对话，并改善代表不足的群体的教育和科学氛围。这个项目解决了分析和数论交界处的几个基本问题。首先，PI在涉及曲线子簇上积分的调和分析中追求连续算子的离散变量的界。这些界限提供了关于定义这些变种的基本丢番图方程的定量分布事实，这使得它们不同于连续的变种。特别是，由于连续技术通常不会在这种情况下继续下去，PI将发展精炼的数论技术来约束几个运算符，包括多线性球面变量、定义在素数上的变量以及更高的余维类似物。特别是，更高的协维度研究应该会开辟新的问题途径，因为在这种情况下人们知之甚少。解决这些问题与离散几何、曲面的格点计数和Falconer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“稀疏界”。稀疏界是对勒贝格空间界的改进，允许人们推导出加权估计。最后，PI计划在算术统计方面推行一项影响深远的计划。这是最近在代数方面得到很大发展的一个领域。PI计划注入傅立叶分析技术，以获得精确的格点计数，这些点阵计数适用于利用代数技术的力量，并将这些界限推向更远。特别是，PI希望获得某些物体的计数，如椭圆曲线，不仅着眼于开发技术，还着眼于以新的方式促进数字理论家和分析师之间的互动。这一奖项反映了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。

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.