The theme of this project is to study invariants coming from symplectic geometry, gauge theory, and quantum algebra and apply these invariants to questions in low-dimensional topology and geometry. A central objective is to understand the topological quantum field theoretic aspects of various Floer homology theories in dimensions (2+1). Goals of the project include the discovery of formulas for the Floer homology of three-manifolds obtained by gluing together manifolds with boundary, the definition of relative Floer invariants for knots and surfaces in the context of contact and symplectic homology theories, and the discovery of topological characterizations of three-manifolds and knots which arise as the boundaries of complex surfaces and curves, respectively. As the invariants studied have already proved to be quite powerful, many possible applications could arise. Among these applications are the study of Dehn surgery problems, smooth concordance and unknotting numbers of knots, and the study of algebraic curves.
The study of three- and four-dimensional spaces, and knotted curves and surfaces within them, is a central task to our understanding of both large and small scale aspects of the universe. Determining the shape of the universe depends upon a mathematical understanding of the possible shapes that could occur and the properties these shapes have. These properties are known as invariants, and the project furthers our understanding of space by the discovery of new invariants and the study of existing invariants. In this pursuit it has been quite fruitful to examine the way in which curves and surfaces can be tied in knots. This kind of knotting is not only relevant to understanding the shape of space, but has recently become significant in the study of DNA. Confined to a small space, long strands of DNA naturally become knotted, and certain processes depend upon an understanding of the complexity of these knots. Applications of this project include effective ways of measuring different types of complexity of knots.
