This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
This research project deals mainly with link homology theories and topological quantum field theories (TQFTs). Link homology theories are algebraically defined invariants of links that generalize and enhance classical invariants such as the Alexander, Jones and quantum sl(n)-polynomials, and they can often be regarded as TQFTs restricted to links in the 3-dimensional space and link cobordisms. There are rank 2 Frobenius extensions that play an important role in such theories, and the Principal Investigator plans to use the techniques developed in the last few years in the area of categorification to find new tangle and link homology theories that are related to rank n-Frobenius extensions for arbitrary n > 2. The project also aims to deepen the understanding of existing link homology theories, including the Khovanov-Rozansky homologies, and to improve the currently known categorifications of the colored Jones polynomial. The novelty of the proposed research lies in a new approach to link homologies via webs and foams and to their extension to cobordisms of knots and links. This new approach also motivates another goal of the project, namely that of constructing extended TQFTs defined on certain cobordisms with seams, also called foams.
The proposed research project concerns the theory of knots and links. This area has provided models and applications to DNA theory, molecular configurations and physics, and has gone through a significant development recently through categorifications of quantum invariants, giving rise to link homologies. The focus of the project is to better perceive the existing link homology theories, to improve some of their features, as well as to find new homology and topological quantum field theories. The Principal Investigator anticipates a better understanding of the quantum sl(n) invariants and of the interplay between knot theory and representation theory. The findings of the study should open new perspectives for applications of methods from the field of homological invariants of knots and links in various branches of mathematics and theoretical physics, including representation theory, category theory, and topological quantum field theory.
