CAREER: Symplectic Weyl Laws, Spectral Invariants, and Beyond

辛几何是经典力学所隐含的几何，而辛几何又是理解宇宙如何运作的核心。动力系统领域研究系统如何随时间演化。这个项目的重点是新的工具，称为谱不变量，它有助于辛形状的研究，并将辛几何与动力系统联系起来。尽管这些不变量在最近被发展和用于解决各种长期存在的猜想，但仍然存在一些谜团，新的视野继续成为焦点。该项目旨在完成与谱不变量相关的基础工作，阐明它们之间的相互关系，证明新的和更精细的结果，探索更高的维度，并研究潜在的新的和重要的应用。协同教育组件的核心是建立一个本科数学研究的数字中心，支持被监禁人员的数学教育，并通过志愿者的努力支持传统课程之外的高中教育。在之前的研究中，PI证明了Weyl定律在2、3和4维的内嵌接触同调、周期Floer同调和链接谱变异体中的辛类似物，从更现代的渐近性中恢复了经典辛不变量。这些想法随后被用于PI的联合工作，以解决各种长期存在的问题，包括已经开放了大约40年的简单性猜想。在目前的项目中，PI旨在探索关于谱不变量的基本问题，证明更精细的Weyl定律，解决各种长期存在的问题，并探索更高维度的自然问题。本项目分为三个部分。在第一部分中，PI将研究谱不变量之间的关系。第二部分是关于次导渐近的两项Weyl定律和猜想。最后一部分涉及应用以及更高维度的新视野。一个潜在应用的例子是为任何维数的所有闭辛流形建立辛填充稳定性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.