Gauge Theory and Invariants of Symplectic Manifolds

这个项目的第一个目标是研究一类被称为广义Seiberg-Witten方程的方程。这些鲜为人知的方程属于被称为规范理论的数学领域，它起源于物理学，描述规范场的动力学，例如电磁场和其他承载自然基本力的场。规范场在纯数学中的重要性源于它们与流形几何的关系。这就引出了这个项目的第二个目标，即应用广义Seiberg-Witten方程的理论来寻找区分不同流形的新方法。首席研究员（PI）对不同研究领域的交叉应用特别感兴趣，例如代数几何（研究由方程描述的流形），辛几何（研究与经典力学相关的流形），拓扑学（研究在变形下保持不变的流形的性质）和弦理论（现代理论物理学的一个分支）。此外，该项目的目标是向广泛的受众，包括研究生和本科生，以及其他领域的研究人员，展示规范理论和几何的一些思想和技术。作为项目的一部分，PI将为学生组织研讨会和迷你课程，并撰写针对非专业人士的规范论说明性笔记。本计画将发展研究三维及四维流形上的广义Seiberg-Witten方程的分析基础。这类方程的例子包括vfa - witten方程和Kapustin-Witten方程，它们有望导致低维流形和结的新的拓扑不变量，以及ADHM Seiberg-Witten方程，它们在定义具有特殊完整度的高维黎曼流形的猜想不变量方面起着重要作用。近年来，在Taubes的开创性工作之后，在这些方程的紧性问题上取得了许多进展。项目的第一个目标是解决描述边界附近解的模空间的逆向问题。这将涉及理解Fueter方程的奇异解，这是Dirac方程的非线性推广，它是广义Seiberg-Witten方程的极限。特别是，目前对这种奇异截面的变形理论和粘接结构知之甚少，在这一方向上取得进展将涉及开发新的分析工具来研究椭圆微分方程的奇异解。PI将应用这一一般理论来定义Calabi-Yau三倍的Pandharipande-Thomas不变量的simsimi模拟。利用ADHM Seiberg-Witten方程的模空间定义权值的嵌入伪全纯曲线的不变量计数。Pandharipande-Thomas不变量的一个辛子解释可能会给mauliki - nekrasov - okounkov - pandharipande猜想带来新的启示，该猜想仍然是Calabi-Yau三倍数列几何的主要开放问题之一。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。

The first objective of this project is to study a class of equations known as generalized Seiberg-Witten equations. These little understood equations belong to the area of mathematics known as gauge theory, which originates from physics and describes the dynamics of gauge fields, such as the electromagnetic field and other fields carrying fundamental forces of nature. The importance of gauge fields in pure mathematics stems from their relationship to the geometry of shapes known as manifolds. This leads to the second objective of this project, which is to apply the theory of generalized Seiberg-Witten equations to find new ways of distinguishing different manifolds from one another. The Principal Investigator (PI) is specifically interested in applications lying at the intersection of different areas of research, such as algebraic geometry (which studies manifolds described by equations), symplectic geometry (which studies manifolds related to classical mechanics), topology (which studies properties of manifolds which remain unchanged under deformations), and string theory (a branch of modern theoretical physics). In addition, the project's goal is to expose a broad audience, including graduate and undergraduate students, as well as researchers in other areas, to some of the ideas and techniques of gauge theory and geometry. As part of the project, the PI will organize seminars and minicourses for students and write expository notes on gauge theory aimed at non-specialists.This project will develop analytic foundations in the study of generalized Seiberg-Witten equations on three- and four-dimensional manifolds. Examples of such equations include the Vafa–Witten and Kapustin–Witten equations, which are expected to lead to new topological invariants of low-dimensional manifolds and knots, and the ADHM Seiberg–Witten equations which play an important role in defining conjectural invariants of higher-dimensional Riemannian manifolds with special holonomy. In recent years, there has been a lot of progress on the compactness problem for these equations, following groundbreaking work of Taubes. The first goal of the project is to solve the converse problem of describing the moduli spaces of solutions near the boundary. This will involve understanding singular solutions to the Fueter equation, a nonlinear generalization of the Dirac equation, which appears as the limit of rescallings of generalized Seiberg–Witten equations. In particular, little is at present known about deformation theory and gluing constructions for such singular sections, and making progress in this direction will involve developing new analytical tools for studying singular solutions of elliptic differential equations. The PI will apply this general theory to define a symplectic analog of the Pandharipande–Thomas invariants of Calabi–Yau threefolds. The invariant counts embedded pseudo-holomorphic curves with weights defined using moduli spaces of the ADHM Seiberg–Witten equations. A symplectic interpretation of the Pandharipande–Thomas invariant is likely to shed a new light on the Maulik–Nekrasov–Okounkov–Pandharipande conjecture, which remains one of the major open problems of enumerative geometry of Calabi–Yau threefolds.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.