The main topic of this project is Floer homology and related invariants in low-dimensional topology. Floer homology is an infinite dimensional version of Morse theory which has been used to construct various invariants of knots, 3-manifolds, 4-manifolds, etc. In turn, these invariants can answer subtle questions about the respective topological objects. One source of invariants with numerous topological applications is Heegaard Floer theory. For example, the Heegaard Floer invariant for knots (called knot Floer homology) is able to detect the genus of a knot. Originally, all the Heegaard Floer invariants were defined in terms of pseudo-holomorphic curves in symmetric products. Recently, the Heegaard Floer invariants of knots, 3-manifolds and 4-manifolds have all been given combinatorial descriptions, based on grid diagrams. One focus of this project is to improve our understanding of these combinatorial descriptions. The PI will also work on finding connections between knot Floer homology and other knot invariants, such as the Khovanov-Rozansky homologies; on extending Heegaard Floer theory to manifolds with corners; and on constructing new Floer-theoretic invariants of three-manifolds using moduli spaces of flat connections.
The proposed research is on topology, an area of mathematics that studies geometric objects such as curved spaces (manifolds), and knots in space. One useful method of studying manifolds and knots is through topological quantum field theories (TQFT's), which are certain toy models used in Mathematical Physics to explore quantum theories about the universe. TQFT's are of interest to topologists because they contain information about the possible shapes of space in various dimensions. An important problem is the classification of these shapes, and this is particularly difficult in four dimensions. Because our macroscopic space-time has four dimensions, the properties of four-dimensional shapes are an essential input for quantum physicists and cosmologists looking for geometric models for the universe. The goal of this project is to study the TQFT's which hold the most promise for our understanding of four-dimensional shapes.
The project concerned various invariants that appear in low dimensional topology, and in particular an invariant called Floer homology. Floer homology is an algebraic way of encoding information from various non-linear differential equations that can be written on a three dimensional space. We focused on the Seiberg-Witten equations, which produce the Seiberg-Witten Floer homology. A similar (and equivalent) theory, called Heegaard Floer homology, can be developed using symplectic geometric techniques. Although Seiberg-Witten Floer homology is associated to three-dimensional spaces, by an indirect route it can give insights into the triangulations of manifolds of dimension five or higher. (A manifold is a space that looks locally like Euclidean space. A triangulation is a decomposition of the space into polyhedra, and gives a simple combinatorial description of the space.) By taking into account an extra symmetry of the Seiberg-Witten equations, the PI developed a new version of Floer homology, and then used it to show the existence of manifolds (in any dimension at least five) that cannot admit any triangulations. This answered a question of Kneser, first posed in 1924. Previously, non-triangulable manifolds were only known to exist in dimension four, due to the work of Casson in the mid 1980s. In a different direction, in joint work with Douglas and Lipshitz, the PI studied the properties of Heegaard Floer homology when one splits a three-dimensional manifold into manifolds with corners. The outcome was a new theory called cornered Heegaard Floer homology. Moreover, in collaboration with computer scientists, the PI explored applications of topology to distributed computing. The main result was to characterize the solvability of a given task by a system of several processors in terms of a question in topology. Previously, such a characterization was available in the wait-free model; that is, assuming that any number of the processors can fail at any given time. The new result is a description that applies to models with restrictions on the possible failures of processors. The PI has disseminated this work through talks at 14 conferences, 13 specialized seminars, and 7 departmental colloquia. He gave six lecture series aimed at graduate students and researchers. The PI has also written three survey articles, with the goal of making recent developments in the field more accessible. Part of the grant was used to support the research of several graduate students: Tye Lidman, Yajing Liu, Jianfeng Lin, and Christopher Scaduto. Lidman solved a conjecture about the infinity flavor of Heegaard Floer homology. Liu computed the Floer homology of manifolds obtained by surgery on two-bridge links. Lin obtained restrictions on which surgeries on knots admit non-cyclic representations to the group SU(2). Scaduto studied the relation between gauge theory (the Yang-Mills equation) and a knot invariant called Khovanov homology.
