Distribution and Analytic Aspects of Cusp Forms

这个项目位于数论和分析的交界处，数论是一门与整数有关的数学学科，分析是研究连续现象的学科。就像我们熟悉的光波和声波一样，连续的物体，如信号或质量在不同几何空间上的分布，可以最好地理解为称为本征函数的更简单的基波的叠加。在具有丰富算术对称集(算术流形)的曲线空间上，这些构建块的作用与它们的几何、动力学和基本的代数结构密切协调，通过尖点形式来发挥作用。本项目将研究非球形尖端形态的极端振荡行为和几何影响，以及尖端形态家族在自然环境空间中的分布。这个奖项还将支持从事PI工作的研究生。自同构形式是分析、表示理论和代数群上的算术的基本构件。从分析的角度来看，尖点形式是包括拉普拉斯算子在内的不变微分算子的联合特征函数，其长期/大规模的分析行为(如其大小)应该反映算术双曲流形上的谱几何和混沌动力学。PI将利用迹公式、数的几何和显式逆来研究非球形尖点形式的这种解析性质。尖点形式自然地出现在族中，在参数的扩展和收缩区域(推广了最初由物理学家提出的韦尔定律)及其分布(包括对称型)中确定族的大小是最有意义的。在这个方向上，将追求非调谐频谱的统一计数语句和界限。该奖项反映了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。

This project lies at the interface of number theory, a mathematical discipline concerned with integers, and analysis, the study of continuous phenomena. Just like the familiar light and sound waves, continuous objects such as signals or mass distributions on spaces of varying geometry can be best understood as superpositions of simpler, fundamental harmonics known as eigenfunctions. On curved spaces with a rich set of arithmetic symmetries (arithmetic manifolds), the role of these building blocks closely attuned to their geometry, dynamics, and the underlying algebraic structure is played by cusp forms. This project will investigate the extreme oscillating behavior and geometric impact of non-spherical cusp forms, and the distribution of families of cusp forms within natural ambient spaces. This award will also support graduate students working with the PI.Automorphic forms are basic building blocks of analysis, representation theory, and arithmetic on algebraic groups. From an analytic perspective, cusp forms are joint eigenfunctions of invariant differential operators including the Laplacian, whose long-term/large-scale analytic behavior (such as their size) should reflect the spectral geometry and chaotic dynamics on arithmetic hyperbolic manifolds. The PI will leverage the trace formula, geometry of numbers, and explicit inversion to study such analytic properties of non-spherical cusp forms. Cusp forms naturally occur in families, and it is of central interest to identify the size of a family in expanding and shrinking regions of adelic parameters (generalizing Weyl's law, originally formulated by physicists) and their distribution including symmetry type. In this direction, uniform counting statements and bounds for non-tempered spectrum will be pursued.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.