Heegaard Floer Homology and Low-Dimensional Topology

拓扑学是研究空间形状的数学基础学科。基本问题是两个空间是否具有相同的形状。此外，如果两个空间有不同的形状，这两个形状有多大的不同？我们能通过某些简单的过程将一个空间转换到另一个空间吗？一个空间是如何被包含在高维空间中的？这些问题是现代数学的基础，对物理、化学和生物学等其他学科也很重要。这个由美国国家科学基金会资助的项目将研究和开发有助于提取空间信息的技术，从而在某些情况下回答这类问题。该项目还旨在使几何学和拓扑学面向广泛的受众，包括数学专业的本科生和其他学科的学者。该项目将纳入加州理工学院的SURF项目，该项目促进了本科生的研究。PI将通过指导研究生和博士后，组织研讨会和会议，在暑期学校和讲习班教授短期课程来帮助扩大这一领域的影响力。本课题主要研究Heegaard flower同调及其在低维拓扑中的应用。eggaard flower同调是由规范论和辛几何中的方法定义的一组不变量。PI将研究Heegaard flower同调与低维拓扑的许多其他方面之间的关系。特别地，PI将研究Gabai的性质R定理在3流形中的零同伦结的推广。通过这样做，PI希望探索Heegaard flower同调与3流形基本群之间的关系。PI计划解决的另一个问题是对3球结进行有限手术的分类。PI还将解决flower同源性和虚拟技术在四维拓扑中的应用，例如，K3表面上结手术的拓扑。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。

Topology is a fundamental discipline of mathematics which studies the shape of spaces. The basic question is whether two spaces have the same shape. Moreover, if two spaces have different shapes, how different are the two shapes? Can we transform one space to another by certain simple processes? How is a space contained in higher dimensional spaces? Such questions are fundamental in modern mathematics, and are important to other disciplines such as physics, chemistry, and biology. This National Science Foundation funded project will study and develop techniques that help extract information about spaces, and thus answer such questions for some cases. This project also aims to make geometry and topology accessible to a broad audience, including undergraduate mathematics majors and scholars from other disciplines. The project will incorporate Caltech's SURF program, which facilitates undergraduate research. The PI will help broaden the influence of this field by mentoring graduate students and postdocs, organizing seminars and conferences, teaching short courses in summer schools and workshops.This project focuses on Heegaard Floer homology and its applications to low-dimensional topology. Heegaard Floer homology is a package of invariants defined via methods in gauge theory and symplectic geometry. The PI will investigate the relationship between Heegaard Floer homology and many other aspects of low-dimensional topology. In particular, the PI will study the generalization of Gabai's Property R theorem to null-homotopic knots in 3-manifolds. By doing so, the PI hopes to explore the relationship between Heegaard Floer homology and fundamental groups of 3-manifolds. Another problem the PI plans to attack is the classification of finite surgeries on knots in the 3-sphere. The PI will also address the applications of Floer homology and virtual techniques to 4-dimensional topology, for example, the topology of knot surgeries on the K3 surface.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.