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，在许多科学领域有具体的应用，如凝聚态物理学，宇宙学，数据分析和生物学。正如人们可以通过听鼓的声音来推断鼓的形状一样，人们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，规范理论是粒子物理学标准模型基本定律的几何语言。该项目的重点是塞伯格-威滕方程，由于其几何性质，提供了一个完美的Vantage位置，以探索拓扑与相邻学科，如双曲几何，谱理论和复分析的相互作用。一个关键的目标是探索新的调查途径在接口与这些领域的数学。为此，该项目也将创造许多研究机会，无论是在本科生和研究生水平。该PI将研究Escheriche Floer同调，与以下主要目标：探索与双曲几何的相互作用，在三维空间中使用的工具，从谱几何，椭圆偏微分方程的理论，和Selberg迹公式;利用指数理论和耦合莫尔斯同调的泛函性质，发展了一般负定配边诱导映射的计算工具;研究与代数几何中经典主题的可能关系，例如黎曼曲面的θ因子的奇异性研究;该奖项反映了NSF的法定使命，并通过使用基金会的智力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.