New Directions in Monopole Floer Homology

拓扑学的目标是确定形状的哪些特征在连续变形下不会改变，具体应用于许多科学领域，如凝聚态物理、宇宙学、数据分析和生物学。正如一个人可以通过听鼓的声音来推断鼓的形状一样，一个人可以通过研究规范理论中自然产生的某些偏微分方程的解来定义3维和4维空间的拓扑不变量，规范理论是一种几何语言，粒子物理标准模型的基本定律就是用它来表述的。该项目侧重于Seiberg-Witten方程，由于其几何性质，为探索拓扑与邻近学科（如双曲几何、光谱理论和复杂分析）的相互作用提供了一个完美的有利位置。一个关键的目标是探索新的研究途径与这些数学领域的接口。为此，该项目还将为本科生和研究生创造许多研究机会。PI将研究单极子Floer同调，主要目标如下：利用光谱几何、椭圆偏微分方程理论和Selberg迹公式等工具，探索单极子Floer同调与双曲几何在三维中的相互作用；利用指数理论和耦合莫尔斯同调的泛函性质，开发了一般负定协映射的计算工具；研究与代数几何中经典主题的可能关系，例如研究黎曼曲面上因子的奇点；并在l空间猜想的精神下，利用Pin(2)-对称来理解具有小花同调的有理同调球的潜在几何特征。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 goal of topology is to identify which features of a shape do not change under a continuous deformation, with concrete applications in many areas of science such as condensed matter physics, cosmology, data analysis, and biology. As one can infer information about the shape of a drum by listening to the way it sounds, one can define topological invariants of spaces of dimension 3 and 4 by studying the solutions of certain partial differential equations naturally arising in gauge theory, the geometric language in which the fundamental laws of the Standard Model of particle physics are formulated. This project focuses on the Seiberg-Witten equations which, because of their geometric nature, provide a perfect vantage point to probe the interactions of topology with neighboring subjects such as hyperbolic geometry, spectral theory, and complex analysis. A key objective is the exploration of new avenues of investigation at the interface with these fields of mathematics. Towards this end, this project will also create many research opportunities both at the undergraduate and graduate level.The PI will study monopole Floer homology, with the following main goals: explore the interactions with hyperbolic geometry in three dimensions using tools from spectral geometry, the theory of elliptic partial differential equations, and the Selberg trace formula; develop computational tools for the maps induced by general negative definite cobordisms using index theory and functoriality properties of coupled Morse homology; investigate possible relations with classical topics in algebraic geometry such as the study of the singularities of the theta divisor of a Riemann surface; and use Pin(2)-symmetry to understand potential geometric characterizations of rational homology spheres with small Floer homology, in the spirit of the L-space conjecture.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.