The goal of this project is to develop concepts in non-commutative (or quantum) algebra as they apply to low-dimensional topology and quantum information theory. The PI plans to study tensor categories, in particular fusion categories and categories arising from quantum groups, that can be used to construct 3-manifold invariants. The PI also plans to study perturbative quantum invariants (due to Vassiliev and Kontsevich) and the related problem of deformation theory of tensor categories. In the area of quantum information theory, the PI plans to study analogues of foundational results in probability theory such as the central limit theorem, as well as possible applications of representation theory and quantum groups.
One of the foundations of modern mathematics and physics is non-commutative algebra, which is the study of abstract formulas in which multiplication does not commute (although addition still does). Non-commutative algebra is the essence of quantum mechanics, and it is also fundamental for understanding symmetry, The purpose of this project is to further apply non-commutative algebra to quantum computation and 3-dimensional topology. Quantum computation is at once an important theoretical elaboration of quantum mechanics and a possible future technology. Topology is the study of knots, links, and abstract geometric spaces. It is closely related to symmetry and it has applications to many other areas of mathematics, as well as to questions in physics and economics and even to the study of DNA.
