Program

The aims of this program will be to achieve the following goals: Promote communication with related disciplines, including the symplectic geometry program in 2009-2010.Lead to new breakthroughs in the subject and find new applications to low dimensional topology (knot theory, three-manifold topology, and smooth four manifold topology).Educate a new generation of graduate students and PhD students in this exciting and rapidly-changing subject. The program will focus on algebraic link homology and Heegaard Floer homology. Khovanov's theory of links is a very young and rapidly developing area drawing on many branches of mathematics. The subject has its roots in representation theory, and it has benefited from its interactions with low dimensional classical and quantum topology and symplectic geometry. In the short period since its birth, link homology has already exhibited the remarkable feature of fusing together many distinct areas of mathematics. We anticipate further connections with hyperbolic geometry, combinatorics, smooth four-manifold topology, string theory, geometric representation theory and the Langlands program. From a different direction, Heegaard Floer homology is an invariant for low-dimensional manifolds whose discovery was inspired by gauge theory and its conjectural connections with symplectic geometry. Although this subject grew out of a different mathematical background from Khovanov's theory, the two subjects are clearly coalescing to give a picture of topological quantum field theories in low-dimensional topology.