Advances in Spectral Theory, Several Complex Variables, and the Geometry of Eigenfunctions of the Laplacian

The project resides at the intersection of the fields of mathematical analysis, partial differential equations, and geometry. Specific questions to be investigated concern the inverse spectral problem, the structure of Laplacian eigenfunctions, and the asymptotic behavior of Bergman kernels. These topics are crucial for understanding how geometric structure influences spectral properties, and are deeply connected to various concepts in mathematics and mathematical physics. The inverse spectral problem investigates how much information about the shape of a (not necessarily round) drum can be obtained solely from its frequencies of vibration. This is a classical question famously popularized by Mark Kac with the phrase `Can one hear the shape of a drum?’ A second set of problems focuses on the shape and the size of the set (`nodal set’) on which an eigenstate of the Laplacian operator vanishes. The size (in the sense of Hausdorff measure) of the nodal set is conjectured to be comparable to the frequency of the eigenstate. Finally, the Bergman kernel is an essential concept in the fields of complex analysis and complex geometry. Bergman kernel approximations arise naturally in mathematical physics, especially in string theory, where they have been proposed as a tool for the search for geometrically well-behaved complex metric structures. The broader impacts of the project contribute to education and diversity. The principal investigator actively supervises graduate students and postdoctoral researchers. Additionally, the PI participates in outreach activities such as organizing math competitions for middle school students and mentoring students from diverse backgrounds through summer research programs. The planned research lies in three main areas. Building on previous work with Steve Zelditch, in which nearly circular ellipses were shown to be spectrally unique among smooth domains, the PI aims to generalize such results to generic ellipses and to domains with constant width. Another direction of interest resides in strong inverse spectral results for generic polygons. Next, recent work by the PI resulted in new explicit upper bounds for the Hausdorff measure of nodal sets of eigenfunctions on compact Riemannian manifolds with Gevrey or quasianalytic regularity. The PI seeks to extend these results to manifolds with boundaries and to improve the current upper bounds using innovative methods. Finally, in collaboration with Hang Xu, the PI investigates asymptotic properties of Bergman kernels. This includes establishing new off-diagonal asymptotics for smooth Kaehler potentials and improving extant upper bounds for Bergman kernels. In addition, the PI will explore convergence properties of the Fefferman expansion for domains with real analytic boundaries. 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.