This research project involves studying 3- and 4-dimensional spaces called manifolds, and geometric structures on them. Understanding these spaces and structures is central to understanding the shape and properties of our universe. For example, Einstein's description of gravity via general relativity relies on mathematical tools for understanding manifolds that are not too different from those under development in this project. Indeed, many of the theories under study have applications and origins in physics. A major goal of the project is to develop methods that might be used to unify several existing and important tools for studying 3- and 4-dimensional manifolds. Such unification would likely reveal interesting connections between disparate fields of mathematics. An important method for understanding higher dimensional spaces is to study lower dimensional spaces that embed into them; much of this research project will be focused on understanding 1-dimensional spaces called knots embedded in 3-dimensional manifolds. Knot theory was initiated as a subfield of topology in the 1880's by the chemist Lord Kelvin, and it has many other interesting applications outside of mathematics -- to the study of how certain enzymes alter the knotting of DNA, for example. One of the project goals is to unify two very important, but prima facie different, tools for studying knots; there is tantalizing conjectural evidence of a concrete relationship between these two theories. Another means of studying 3- and 4-dimensional manifolds is by understanding the sorts of geometric structures they support. Considerable effort in this project will go into understanding contact structures on 3-dimensional manifolds. In addition to their applications in mathematics, contact structures are important in classical mechanics, thermodynamics, dynamical systems, and in the study of liquid crystals. This project also contains several explicit directions for student research, at both undergraduate and graduate levels, and will support the PI's teaching and mentoring activities. In addition, the project will serve the broader community by creating a Wiki repository of open problems, with background and context. The PI will also use the funds from this award to start a Boston-area graduate seminar, and to participate in math outreach for the general public at the Cambridge Science Festival.
Floer theory has revolutionized the study of the topology and geometry in low dimensions. Many of these Floer theories appear to encode the same information, indicating deep connections between fields like symplectic geometry and gauge theory. These and connections between Floer theory and link invariants from representation theory have led to spectacular results in low-dimensional topology, such as proof of the Weinstein conjecture. Despite some progress in understanding these connections, there is not yet a simple, unifying explanation for them. Indeed, a fundamental open problem, and a distant lodestar for part of the research project, is to axiomatize Floer theory. One approach involves developing invariants of bordered 3-manifolds. The bordered monopole Floer theory under development in this project is novel in that it may provide methods for computing invariants of 4-manifolds as well. A complementary goal of this project is to elucidate links between Floer theory and Khovanov homology, an invariant of links motivated by representation theory. Such connections were instrumental, for example, in establishing that Khovanov homology detects the unknot. This project investigates a new approach to proving a long-standing conjecture relating Khovanov homology with knot Floer homology, which would imply, among other things, that Khovanov homology detects the figure eight, trefoil, and Hopf link as well. The project also investigates a novel approach for using actions of quantum cohomology on Floer theory to study the rich theory of Heegaard splittings of 3-manifolds. This has not been done before, despite the role Heegaard splittings play in the constructions of some Floer theories. Finally, the research project will employ newly-developed gauge-theoretic contact invariants to provide fresh insights into Legendrian knots and Lagrangian concordance, and to establish hitherto unexplored connections between contact geometry and the fundamental group.
