Questions at the Interface of Analysis and Number Theory

调和分析和数论是数学的基本领域，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（如函数）分解为更容易理解的部分。 这一领域的美丽之处在于，这些作品往往很简单，但却准确地代表了整体。 数论涉及到关于整数的看似简单的陈述，容易测试，但往往难以证明。 虽然看似不同，分析和数论共享许多相互作用。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。 这个项目探讨了这两个领域的接口的各种问题。 特别是，PI将在分析中考虑算子的离散变量，这些变量在医学成像和宇宙学等领域中具有应用。 为了分析这些算子，连续技术往往失败，人们必须开发适合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，更清晰的分析和更广泛的联系。 PI还计划将傅立叶分析（一种时频域的基本分解方法，如用于理解波浪的方法）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。 作为一个更广泛的影响，PI将引发分析师和数论家之间新的数学对话，并改善教育和科学氛围的代表性不足的群体。这个项目解决了分析和数论接口的几个基本问题。 首先，PI追求调和分析中连续算子的离散变量的界，这些算子涉及在弯曲子簇上的积分。 这些边界提供了关于定义这些变量的基本丢番图方程的定量分布事实，这使得它们不同于它们的连续对应物。 特别是，由于连续技术通常不会在这种情况下延续，PI将开发改进的数论技术来约束几个运算符，包括多线性球形变体、在素数上定义的变体以及更高的余维类似物。 特别是，更高的余维研究应该开辟新的途径的问题，因为很少有人知道在这种情况下。解决这些问题与离散几何、曲面的格点计数和法尔科纳距离猜想有关。 在另一系列的问题中，PI将追求连续和离散算子的“稀疏边界”。 稀疏边界是勒贝格空间边界的细化，允许推导加权估计。 最后，PI计划在算术统计方面开展一项意义深远的计划。 这是一个领域大大发展的代数方面最近。PI计划注入傅立叶分析技术，以获得精确的格点计数，这些格点计数可以利用代数技术的力量，并进一步推动这些边界。 特别是，PI希望获得椭圆曲线等特定对象的计数，不仅着眼于技术的开发，还着眼于以新的方式促进数论家和分析家之间的互动。该奖项反映了NSF的法定使命，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。

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.