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.
