Ergodicity and the Number of Nodal Domains of Eigenfunctions of the Laplacian
拉普拉斯本征函数的遍历性和节点域数
基本信息
- 批准号:1900993
- 负责人:
- 金额:$ 15万
- 依托单位:
- 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:2019-07-15 至 2020-09-30
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The main research objective of this project is to investigate mathematically a newly discovered link between two seemingly unrelated physical phenomena. For instance, there are two physical experiments one can perform with a bounded, smooth planar lamina. Firstly, one can hit a billiard ball and watch its trajectory over a long time as it bounces off the boundary of the lamina. Secondly, one can resonate the lamina and observe the modes of vibration as the frequency increases. One of the oldest works regarding the latter experiment includes Chladni's experiment in the 18th century. The PI and collaborators recently discovered that there is an unexpected connection between the types of trajectories one observes, and the geometry of patterns one obtains as a result of vibration. In modern language, this is a study of the relation between classical dynamics and corresponding quantum dynamics. The project aims to further explore this newly found connection in various setup to expand our knowledge about the impact of the presence of chaos in classical dynamics to the corresponding quantum dynamics. This research involves training upper class undergraduate students and graduate students, and developing a topic course for them.To be specific, the PI is interested in the impact of ergodicity of geodesic flow on the geometry of nodal sets of eigenfunctions of the Laplace-Beltrami operator. The PI proposes to investigate the geometry of nodal sets in two contrasting cases: hyperbolic 3 manifolds with a cusp and circle bundles over closed surfaces endowed with a metric that is invariant under the circular action. The geodesic flow is always ergodic in the former examples, and never ergodic in the latter examples. The complexity of the nodal set will be measured by the zeroth and the first Betti numbers. The PI also proposes to study the long-standing problem regarding the existence of embedded eigenvalues of Hodge Laplacian. This is the first step to be done in order to understand the interplay between ergodicity and eigenforms of Hodge Laplacian.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.
这个项目的主要研究目标是对两个看似无关的物理现象之间的一种新发现的联系进行数学研究。例如,有两个物理实验可以用一个有边界的光滑的平面薄板来进行。首先,人们可以击打台球,并在很长一段时间内观察它从板层边界反弹的轨迹。其次,人们可以使薄板共振,并观察到振动模式随着频率的增加而增加。关于后一种实验的最古老的著作之一包括18世纪的切拉德尼的实验。PI和合作者最近发现,在人们观察到的轨迹类型和作为振动结果而获得的图案的几何形状之间存在着意想不到的联系。在现代语言中,这是对经典动力学和相应的量子动力学之间关系的研究。该项目的目的是在不同的设置下进一步探索这种新发现的联系,以扩大我们关于经典动力学中混沌的存在对相应量子动力学的影响的知识。这项研究包括培训高年级本科生和研究生,并为他们开发一门专题课程。具体地说,PI感兴趣的是测地线流的遍历性对Laplace-Beltrami算子的特征函数节点集几何的影响。PI建议在两种不同的情况下研究节点集的几何:带尖点的双曲3流形和具有在圆作用下不变的度量的闭曲面上的圆丛。测地线流在前面的例子中总是遍历的,在后一个例子中从来不是遍历的。节点集的复杂性将通过第零个和第一个Betti数来衡量。PI还建议研究关于Hodge Laplace算子嵌入特征值的存在性的长期存在的问题。这是为了理解霍奇·拉普拉斯的遍历性和特征形式之间的相互作用而要做的第一步。这一奖项反映了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。
项目成果
期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Topology of the Nodal Set of Random Equivariant Spherical Harmonics on ?3
?3 上随机等变球谐函数节点集的拓扑
- DOI:10.1093/imrn/rnz348
- 发表时间:2020
- 期刊:
- 影响因子:1
- 作者:Jung, Junehyuk;Zelditch, Steve
- 通讯作者:Zelditch, Steve
Asymptotic trace formula for the Hecke operators
Hecke 算子的渐近迹公式
- DOI:10.1007/s00208-020-02054-w
- 发表时间:2020
- 期刊:
- 影响因子:1.4
- 作者:Jung, Junehyuk;Talebizadeh Sardari, Naser
- 通讯作者:Talebizadeh Sardari, Naser
Boundedness of the number of nodal domains for eigenfunctions of generic Kaluza–Klein 3-folds
泛型 Kaluza-Klein 3 倍本征函数的节点域数量有界性
- DOI:10.5802/aif.3329
- 发表时间:2020
- 期刊:
- 影响因子:0
- 作者:Jung, Junehyuk;Zelditch, Steve
- 通讯作者:Zelditch, Steve
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
数据更新时间:{{ journalArticles.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ monograph.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ sciAawards.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ conferencePapers.updateTime }}
{{ item.title }}
- 作者:
{{ item.author }}
数据更新时间:{{ patent.updateTime }}
Junehyuk Jung其他文献
On the sparsity of positive-definite automorphic forms within a family
- DOI:
10.1007/s11854-016-0017-9 - 发表时间:
2016-08-25 - 期刊:
- 影响因子:0.900
- 作者:
Junehyuk Jung;Sug Woo Shin - 通讯作者:
Sug Woo Shin
Sign changes of the Eisenstein series on the critical line
临界线上爱森斯坦级数的符号变化
- DOI:
- 发表时间:
2016 - 期刊:
- 影响因子:0
- 作者:
Junehyuk Jung;M. Young - 通讯作者:
M. Young
$2$-nodal domain theorems for higher dimensional circle bundles
高维圆丛的 $2$-节点域定理
- DOI:
- 发表时间:
2022 - 期刊:
- 影响因子:0
- 作者:
Junehyuk Jung;S. Zelditch - 通讯作者:
S. Zelditch
Sharp bounds for the intersection of nodal lines with certain curves
节点线与某些曲线相交的尖锐边界
- DOI:
10.4171/jems/433 - 发表时间:
2011 - 期刊:
- 影响因子:0
- 作者:
Junehyuk Jung - 通讯作者:
Junehyuk Jung
Bounding the number of nodal domains of eigenfunctions without singular points on the square
限制正方形上没有奇异点的特征函数的节点域数量
- DOI:
- 发表时间:
2017 - 期刊:
- 影响因子:1
- 作者:
Junehyuk Jung - 通讯作者:
Junehyuk Jung
Junehyuk Jung的其他文献
{{
item.title }}
{{ item.translation_title }}
- DOI:
{{ item.doi }} - 发表时间:
{{ item.publish_year }} - 期刊:
- 影响因子:{{ item.factor }}
- 作者:
{{ item.authors }} - 通讯作者:
{{ item.author }}
{{ truncateString('Junehyuk Jung', 18)}}的其他基金
Ergodicity and the Number of Nodal Domains of Eigenfunctions of the Laplacian
拉普拉斯本征函数的遍历性和节点域数
- 批准号:
2050123 - 财政年份:2020
- 资助金额:
$ 15万 - 项目类别:
Standard Grant
相似国自然基金
关于群上的短零和序列及其cross number的研究
- 批准号:11501561
- 批准年份:2015
- 资助金额:18.0 万元
- 项目类别:青年科学基金项目
相似海外基金
Dynamical Approaches to Number Theory and Additive Combinatorics
数论和加法组合学的动态方法
- 批准号:
EP/Y014030/1 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Research Grant
Conference: Pittsburgh Links among Analysis and Number Theory (PLANT)
会议:匹兹堡分析与数论之间的联系 (PLANT)
- 批准号:
2334874 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Standard Grant
REU Site: Computational Number Theory
REU 网站:计算数论
- 批准号:
2349174 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Continuing Grant
Analytic Number Theory at the Interface
界面上的解析数论
- 批准号:
2401106 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Continuing Grant
Conference: Solvable Lattice Models, Number Theory and Combinatorics
会议:可解格子模型、数论和组合学
- 批准号:
2401464 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Standard Grant
Class numbers and discriminants: algebraic and analytic number theory meet
类数和判别式:代数和解析数论的结合
- 批准号:
DP240100186 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Discovery Projects
Effect of Reynolds number on drag reduction: from near-wall cycle to large-scale motions.
雷诺数对减阻的影响:从近壁循环到大规模运动。
- 批准号:
2345157 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Standard Grant
Conference: Southern Regional Number Theory Conference
会议:南方区域数论会议
- 批准号:
2341365 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Standard Grant
A1-Homotopy Theory and Applications to Enumerative Geometry and Number Theory
A1-同伦理论及其在枚举几何和数论中的应用
- 批准号:
2405191 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Standard Grant
Conference: Comparative Prime Number Theory Symposium
会议:比较素数论研讨会
- 批准号:
2411537 - 财政年份:2024
- 资助金额:
$ 15万 - 项目类别:
Standard Grant