This project uses holomorphic curves to study questions in the intersection of symplectic geometry and low-dimensional topology. One main goal is developing an algebraic framework for the relative version of Symplectic Field Theory, with applications to contact geometry and knot theory; significant partial progress has recently been made by the Principal Investigator and other researchers. A related goal is studying the smooth topology of low-dimensional objects (e.g., knots and three- and four-manifolds) via the Symplectic Field Theory of cotangent bundles. The Principal Investigator has previously used this strategy to introduce a knot invariant called knot contact homology, which has partially understood connections to string topology. This project will develop the theory of knot contact homology by studying its generalization in Symplectic Field Theory and extending it to invariants of low-dimensional manifolds. Possible applications include relations to similar homological invariants such as Heegaard Floer homology and Khovanov homology, and new approaches to problems in low-dimensional topology such as knot concordance and distinguishing smooth structures on manifolds.
Low-dimensional topology, or the study of shapes in three and four dimensions, is a classical subject of paramount importance in mathematics and of natural interest to other sciences, especially physics. In recent years, many fundamental and longstanding open problems in low-dimensional topology have proven remarkably amenable to new techniques from a different mathematical field, symplectic geometry. Much of this progress has involved the study of holomorphic curves in symplectic manifolds, which has close ties to string theory in physics. The Principal Investigator will use his past work on holomorphic curves within the framework of Symplectic Field Theory to develop new invariants of topological structures such as knots and three-dimensional spaces. This will likely have interesting applications to low-dimensional topology, including a long-term goal to resolve the still-unsolved smooth four-dimensional Poincare conjecture using symplectic techniques. The Principal Investigator will also pursue several educational endeavors, including the development of a new general-interest course for undergraduates, introducing methods of mathematical reasoning through analysis of patterns in nature, the arts, and everyday life.
