The circle method, character sums, and sieves: Applications to number theory and harmonic analysis

数学中的许多问题都与弯曲物体的行为有关。在一个离散的环境中，这些问题中的某些可以追溯到古代，并采取寻求丢番图方程的积分解的形式，这也可以被视为位于“弯曲”变种上的点。在连续的情况下，这样的问题可以采取这样的形式：当一个算子在一个弯曲的子流形上对一个函数求平均值时，它是如何变换这个函数的。这一建议考虑了数论和调和分析中涉及曲率物体的一类广泛问题。PI将在圆法、字符和和筛子方面发展方法学上的进步，导致应用于广泛的数论问题，这些问题可以解释为丢番图方程系统的计数解决方案。此外，PI将研究Radon型的多项式Carleson算子，曲率的存在为其带来了新的困难和美丽的新结构。最后，PI将应用数论和解析方法来证明涉及曲率的离散算术算子的算子界。所有这些问题的核心在于对振荡积分及其离散特征和的研究。在整个研究项目期间，PI将向更广泛的社区推广数学，并鼓励代表性不足的群体参与。

Many questions in mathematics concern the behavior of curved objects. In a discrete setting, certain of these questions date to antiquity and take the form of seeking integral solutions to Diophantine equations, which can also be viewed as points lying on a "curved" variety. In a continuous setting, such questions can take the form of asking how an operator transforms a function when it averages the function over a curved submanifold. This proposal considers a broad class of questions in number theory and harmonic analysis involving objects with curvature.The PI will develop methodological advances for the circle method, character sums, and sieves, leading to applications to a wide array of number theoretic questions that can be interpreted as counting solutions to systems of Diophantine equations. In addition, the PI will investigate polynomial Carleson operators of Radon type, for which the presence of curvature introduces both new difficulties and beautiful new structure. Finally, the PI will apply number theoretic and analytic methods to prove operator bounds for discrete arithmetic operators involving curvature. At the heart of all these problems lies the investigation of oscillatory integrals and their discrete cousins character sums. Throughout the duration of the research program, the PI will promote mathematics to the broader community and encourage the participation of underrepresented groups.