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

数学中的许多问题都与曲线物体的行为有关。在离散的环境中，这些问题中的某些可以追溯到古代，并以寻求丢番图方程的整体解的形式出现，该方程也可以被视为位于“曲线”变体上的点。在连续的设置中，这样的问题可以采取这样的形式，即询问当一个函数在曲线子流形上求平均时，算子如何变换该函数。这项建议考虑了涉及曲率对象的数论和调和分析中的一大类问题。PI将发展圆法、特征和和筛子的方法学进步，导致广泛的数论问题的应用，这些问题可以解释为丢番图方程组的计数解。此外，PI还将研究Radon型多项式Carleson算子，对于这种算子，曲率的存在既带来了新的困难，又带来了美丽的新结构。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和它们的离散同类特征和的研究。在整个研究计划期间，国际数学协会将向更广泛的社区推广数学，并鼓励代表不足的群体参与。

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.