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Distribution and Analytic Aspects of Cusp Forms

尖点形式的分布和分析方面

基本信息

批准号：
1903301
负责人：
Djordje Milicevic
金额：
$ 20.1万
依托单位：
Bryn Mawr College
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1903301&HistoricalAwards=false
关键词：
Distribution Analytic Aspects Cusp Forms

项目摘要

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.

这个项目是在接口的数论，一个数学学科有关的整数，分析，研究连续现象。就像我们熟悉的光波和声波一样，连续的物体，如信号或不同几何空间上的质量分布，可以最好地理解为更简单的基本谐波的叠加，称为本征函数。在具有丰富算术对称集的弯曲空间（算术流形）上，这些与几何、动力学和基本代数结构紧密协调的构建块的作用由尖点形式发挥。这个项目将研究极端的振荡行为和非球形尖点形式的几何影响，以及自然环境空间中尖点形式家族的分布。这个奖项也将支持研究生与PI合作。自守形式是分析，表示论和代数群算术的基本构建模块。从分析的角度来看，尖点形式是不变微分算子（包括拉普拉斯算子）的联合特征函数，其长期/大尺度分析行为（如它们的大小）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，几何数字和显式反演来研究非球形尖点形式的分析性质。尖点形式自然地出现在族中，并且确定族在扩展和收缩区域的adelic参数（推广Weyl定律，最初由物理学家制定）及其分布（包括对称类型）中的大小是核心兴趣。这个奖项反映了NSF的法定使命，并被认为是值得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的支持。