LEAPS-MPS: Investigation on Spectral Geometry of Steklov Eigenvalues

早在中世纪，钟匠就知道如何在地上敲钟，然后把它举到钟楼上，这样就能发现看不见的裂缝。从那时起，声音和几何结构之间的关系一直是数学中感兴趣的中心话题，在更广泛的科学界有着重要的应用。在这种情况下，Steklov本征值扮演着重要的角色，例如，在医学成像的电阻抗层析成像中，以及在物理中线性水波的研究中，Steklov特征值是必不可少的。在这个项目中，PI将揭示Steklov特征值和从中导出它们的基本几何之间的新关系。该项目的资金还将通过辅导和会议组织，支持PI努力使学生和年轻学者能够接触和受益于这项研究，并促进未被充分代表的群体参与数学。该项目涉及与Steklov本征值的谱几何相关的两个研究方向。在一个方向上，PI将研究Steklov特征值和特征函数的一般性质，并将研究区域扰动、微分形式及其在等分布定理中的应用。在另一个方向，PI将通过研究折叠流形来研究光滑度量空间上的Steklov特征值。作为加州州立大学长滩分校特别会议和杰出讲座系列的组织者，国际数学联合会将促进早期职业、女性和其他在数学领域代表性不足的群体的参与。PI还将为本科生组织数学专业的职业小组讨论，并为高中生组织“海滩数学日”。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。

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.