CAREER: Symplectic Weyl Laws, Spectral Invariants, and Beyond

University Of Maryland, College Park, College Park MD

Investigators

Abstract

Symplectic geometry is the geometry implied by classical mechanics, which in turn is central to understanding how our universe works. The field of dynamical systems studies how systems evolve in time. This project’s focus is on new tools, called spectral invariants, that facilitate a study of symplectic shapes and connect symplectic geometry to dynamical systems. Although these invariants have been developed and used to resolve various longstanding conjectures in recent times, several mysteries remain, and new horizons continue to come into focus. The project aims to complete foundational work related to spectral invariants, clarify how they relate to each other, prove new and more refined results, explore higher dimensions, and study potential new and important applications. A synergistic educational component is centered around building a digital hub for undergraduate mathematical research, supporting mathematics education of incarcerated individuals, and supporting high school education beyond the traditional curriculum through volunteer efforts. In previous research, the PI proved symplectic analogs of Weyl laws for embedded contact homology, periodic Floer homology, and link spectral variants in dimensions 2, 3, and 4, recovering classical symplectic invariants from the asymptotics of more modern ones. These ideas were then used in joint work of the PI settling various longstanding questions, including the Simplicity Conjecture that had been open for about forty years. In the current project the PI aims to explore foundational questions about the spectral invariants, prove more refined Weyl laws, settle various longstanding problems, and explore natural questions in higher dimensions. The project is divided into three parts. In the first part, the PI will study the relationships between the spectral invariants. The second part is about two-term Weyl laws and conjectures regarding the subleading asymptotics. The final part involves applications as well as new horizons in higher dimensions. An example of a potential application is establishing symplectic packing stability for all closed symplectic manifolds of any dimension. 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.