Questions at the Interface of Analysis and Number Theory

谐波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。谐波分析包括将一个数学对象（如函数）分解成更容易理解的部分。这个领域的美妙之处在于，这些作品往往很简单，但却准确地代表了整体。数论包含了关于整数的看似简单的陈述，易于检验，但往往难以证明。虽然看起来完全不同，但分析学和数论有很多共同之处。例如，人们可以使用复杂函数的复杂分析来回答关于素数的基本问题。这个项目探讨了这两个领域交界的各种问题。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享有应用。为了分析这些运算符，连续技术经常失败，因此必须开发适合于解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（傅里叶分析是一种时频域的基本分解方法，就像用来理解波的方法一样）引入算数统计学这一新兴领域。在这里，她试图提供各种各样的算术兴趣对象的清晰计数，例如密码学中使用的椭圆曲线。作为一个更广泛的影响，PI将激发分析师和数论学家之间的新的数学对话，并改善未被充分代表的群体的教育和科学氛围。这个项目在分析和数论的界面上解决了几个基本问题。首先，PI追求调和分析中连续算子的离散变异体的界，这些变异体涉及对一个弯曲子变异体的积分。这些边界提供了关于定义这些变化的底层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。特别是，由于连续技术通常不会在这种情况下延续，PI将开发精细的数论技术来绑定几个算子，包括多线性球面变异体，定义在素数上的变异体，以及更高的协维类似物。特别是，更高的共维研究应该开辟新的问题途径，因为在这种情况下所知甚少。解决这些问题与离散几何、曲面的点阵计数和Falconer的距离猜想有关。在另一系列问题中，PI将对连续和离散算子追求“稀疏边界”。稀疏边界是勒贝格空间边界的一种改进，它允许推导加权估计。最后，PI计划在算术统计方面进行深远的计划。这是最近在代数方面得到很大发展的一个领域。PI计划注入傅里叶分析技术来获得精确的晶格点计数，这些点计数可以利用代数技术的力量，并进一步推动这些界限。特别是，PI希望获得某些物体（如椭圆曲线）的计数，不仅着眼于发展技术，而且还以新的方式促进数论学家和分析师之间的互动。该奖项反映了美国国家科学基金会的法定使命，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。

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.