CAREER: Symplectic Weyl Laws, Spectral Invariants, and Beyond

辛几何是经典力学所隐含的几何，而这反过来又是理解我们的宇宙如何运行的核心。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，它促进了辛形状的研究，并将辛几何与动力系统联系起来。虽然这些不变量最近被开发并用于解决各种长期存在的猜想，但仍有几个谜团，新的视界继续成为焦点。该项目的目标是完成与光谱不变量相关的基础工作，阐明它们之间的关系，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。协同教育部分围绕建立本科数学研究的数字中心，支持被监禁个人的数学教育，并通过志愿者努力支持传统课程之外的高中教育。在之前的研究中，PI证明了嵌入接触同调、周期Floer同调和维2、3和4的链谱变量的Weyl定律的辛类似，从更现代的辛不变量中恢复经典的辛不变量。这些想法随后被用于国际和平研究所的联合工作，以解决各种长期存在的问题，包括已经存在了大约40年的简单性猜测。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，并在更高的维度上探索自然问题。该项目分为三个部分。在第一部分中，PI将研究谱不变量之间的关系。第二部分是关于二项Weyl定律和关于次偏渐近的猜想。最后一部分涉及应用以及更高维度的新地平线。一个潜在的应用例子是为所有任何维度的闭辛流形建立辛包稳定性。这一奖项反映了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。

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.