CAREER: Heegaard Floer homology and low-dimensional topology

本计画研究Heegaard Floer同调及其在低维拓扑中的应用。Heegaard Floer同调是通过规范理论和辛几何中的方法定义的一组不变量。PI将研究Heegaard Floer同调与低维拓扑的许多其他方面之间的关系。更具体地说，PI将研究Heegaard Floer同源性的地理和植物学问题。通过这样做，PI希望回答有关Dehn手术和Khovanov同源性的问题。PI还将讨论Floer同调在四维拓扑中的应用，例如K3曲面上的结手术拓扑。拓扑是研究空间形状的数学基本学科。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？我们能否通过某些简单的过程将一个空间转换成另一个空间？本项目研究的技术提供了一种提取空间信息的方法，从而在某些情况下回答了以前的问题。该项目还旨在使几何和拓扑学能够被广泛的受众所接受，包括本科数学专业和其他学科的学者。PI将根据本项目研究的主题设计新课程。此外，这个项目将纳入加州理工学院的SURF计划，这为本科生提供了一个做研究的机会。PI还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中教授短期课程来扩大这一领域的影响力。

This project studies 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. More specifically, the PI will study the geography and botany problems of Heegaard Floer homology. By doing so, the PI hopes to answer questions about Dehn surgeries and Khovanov homology. The PI will also address the applications of Floer homology to 4-dimensional topology, for example, the topology of knot surgeries on the K3 surface.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 these two shapes are? Can we transform one space to another by certain simple processes? The techniques studied in this project give a way to extract information about spaces, thus answer the previous questions in some cases. This project also aims to make geometry and topology accessible to a broad audience, including undergraduate math majors and scholars from other disciplines. The PI will design new courses on the topics studied in this project. In addition, this project will incorporate Caltech's SURF program, which provides a chance to undergraduate students to do research. The PI will also broaden the influence of this field by mentoring graduate students and postdoctors, organizing seminars and conferences, teaching short courses in summer schools and workshops.