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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.

这个项目是数论和分析的交汇处，数论是一门研究整数的数学学科，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，如信号或变化几何空间上的质量分布，可以最好地理解为被称为本征函数的更简单的基本谐波的叠加。在具有丰富的算术对称性（算术流形）的弯曲空间上，这些构建块与其几何、动力学和底层代数结构密切协调的作用由尖点形式发挥。该项目将研究非球面尖形的极端振荡行为和几何影响，以及尖形族在自然环境空间中的分布。该奖项还将支持与PI一起工作的研究生。自同构形式是代数群的分析、表示理论和算术的基本组成部分。从解析的角度来看，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/大规模解析行为（如其大小）应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、数字几何和显式反演来研究非球面尖头形式的解析性质。尖点形式自然地出现在家族中，在阿德尔参数（推广Weyl定律，最初由物理学家提出）的扩张和缩小区域以及它们的分布（包括对称类型）中确定家族的大小是中心兴趣。在这个方向上，将追求非回火谱的统一计数语句和界。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。