Heegaard Floer Homology and Low-Dimensional Topology

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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.