LEAPS-MPS: Investigation on Spectral Geometry of Steklov Eigenvalues

LEAPS-MPS：Steklov 特征值的谱几何研究

• 批准号: 2316620
• 负责人: Lihan Wang
• 金额: $ 25万
• 依托单位: California State University-Long Beach Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316620&HistoricalAwards=false
• 关键词: LEAPS; MPS; Investigation; Spectral; Geometry; LEAPS; MPS; Investigation; Spectral; Geometry

Already in the Middle Ages bell makers knew how to detect invisible cracks by sounding a bell on the ground before lifting it up to the belfry. Since then, the relations between sounds and geometric structures have been a central topic of interest in mathematics with important applications in the broader scientific community. Steklov eigenvalues play an important role in this context and, for example, are essential in electrical impedance tomography for medical imaging, and in the study of linear water waves in physics. In this project, the PI will reveal new relations between Steklov eigenvalues and the underlying geometry from which they are derived. Funds for the project will also support the PI’s efforts to make the research accessible and beneficial to students and young scholars and to promote the participation of underrepresented groups in mathematics, through mentoring and conference organization.This project involves two research directions tied to spectral geometry of Steklov eigenvalues. In one direction, the PI will study generic properties of Steklov eigenvalues and eigenfunctions and will investigate domain perturbations, differential forms and their applications to equidistribution theorems. In another direction, the PI will investigate Steklov eigenvalues on smooth metric spaces by studying collapsing manifolds. The PI will promote the participation of early career, female and other underrepresented group in mathematics as organizer of AMS special sessions and distinguished lectures series at Cal State, Long Beach. The PI will also organize career panels for undergraduate math majors, and a “Math Day at the Beach” for high school students.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.

在中世纪，贝尔制造商已经知道如何通过在地面上发出铃铛，然后将其提升到钟楼，从而检测到无形的裂缝。从那时起，声音与几何结构之间的关系一直是在更广泛的科学界中具有重要应用的数学感兴趣的核心话题。 Steklov特征值在这种情况下起着重要的作用，例如，在医学成像的电阻抗中以及对物理学线性水波的研究中至关重要。在这个项目中，PI将揭示Steklov特征值与它们得出的基本几何形状之间的新关系。该项目的资金还将支持PI的努力，以使研究对学生和年轻学者有益，并通过心理和会议组织来促进代表性不足的小组参与数学。该项目涉及与Steklov Eigenvalues的光谱几何学相关的两个研究方向。在一个方向上，PI将研究Steklov特征值和特征功能的通用特性，并将研究域的扰动，差异形式及其在等均分配定理中的应用。在另一个方向上，PI将通过研究崩溃歧管来研究平滑度量空间上的steklov特征值。 PI将促进早期职业生涯，女性和其他代表性不足的小组在数学领域，作为AMS特殊会议和杰出讲座系列的组织者，在长滩加州州立大学。 PI还将为本科数学专业的职业小组组织职业小组，并为高中生举办“在海滩上的数学日”。该奖项反映了NSF的法定任务，并使用基金会的知识分子优点和更广泛的影响评估审查标准，被认为是通过评估来获得的支持。