Link Floer Homology and Kleinian Groups

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A link is a collection of disjoint circles that may be linked together embedded in a space of dimension three. One of the main topics in low-dimensional topology is to study the topological and geometric properties of the link, the three-dimensional space, and some four-dimensional spaces bounded by the three-space. This project aims to deepen understanding of these mathematical structures. The first part of the research concentrates on the study of three-manifolds obtained from links via the so-called "Dehn surgery" operation and the family of links appearing in algebraic geometry. Results are anticipated to advance the understanding of the complexity of three-manifolds and algebraic singularities in algebraic geometry. It will also provide topics that are suitable for undergraduate students' research. The second part of the research focuses on the topology, geometry, and dynamics of hyperbolic manifolds, which are important examples of Gromov hyperbolic spaces, negatively curved Hadamard manifolds, and symmetric spaces of non-compact type.The research consists of four specific projects about links and hyperbolic manifolds. The first aims to understand the possible obstructions for surgeries on 2-component links in the three-sphere. It focuses on the possibility of finding an infinite family of integer homology spheres that cannot be obtained by surgeries on 2-component links in the three-sphere. The second project is to understand the link Floer chain complex of algebraic links coming from the singularities of algebraic curves in the complex plane and provide potential applications in low dimensional topology. The third project is to study discrete isometry subgroups acting on hyperbolic spaces with small critical exponents and generalize the structure theorem for hyperbolic manifolds to negatively curved Hadamard manifolds. The fourth project concerns a counting question in hyperbolic manifolds, with the goal of determining whether the classical Bowen-Margulis measure and the spectral gap converge for a strongly convergent sequence of hyperbolic manifolds.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.