Project Objective

Our collaboration will harness the interplay between low-dimensional geometry, representation theory, and higher-dimensional gauge theory. Through this coordinated effort we expect to see substantial progress on the problem of categorifying 3-manifold invariants. Our team will capitalize on recent breakthroughs in theoretical physics and higher representation theory that have created new possibilities for significant progress on this problem. Among the techniques to be employed include:

• fivebrane compactifications to provide a universal description of various old and new homological invariants of 3-manifolds,
• the use of infinity categories for defining tensor products of higher representations of quantum groups, and
• the theory of Hopfological algebra for categorifications at roots of unity, as well as recent work on odd link homology theory and categorifications of Habiro's universal invariant.

Perhaps more importantly, we expect that the interaction between these proposed approaches will produce fruitful tools and techniques that extend far beyond these considerations.