Quantum low dimensional topology

The whole theory has been, to a great extent, inspired by ideas that arose in theoretical physics. Among the relevant areas of physics are the theory of exactly solvable models of statistical mechanics, the quantum inverse scattering method,the quantum theory of angular momentum, 2-dimensional conformal field theory,etc. The development of this subject shows once more that physics and mathematics intercommunicate and influence each other … - Vladimir G. Turaev

Quantum invariants, founded around 1980 on the ideas of quantum groups and quantum topology, have grown into a rich area of mathematics through the work of mathematicians such as M. Atiyah, L. Kauffman, N. Reshetikhin, V. Turaev, and others. Key developments include the Jones polynomial, introduced by V. Jones in 1984, which revolutionized knot theory, and its extension to 3-manifolds by Witten in 1988. Reshetikhin and Turaev then constructed invariants satisfying topological quantum field theory properties, and the Turaev-Viro invariants introduced in the early 1990s provide a combinatorial framework for computing 3-manifold invariants.

To learn more, see the references below—and keep reading to explore both the projects I have worked on and the exciting directions I plan to pursue next.

Project 1: Seifert fibered 3-manifolds and Turaev-Viro invariants volume conjecture

We study the large $r$ asymptotic behaviour of the Turaev-Viro invariants of oriented Seifert fibered 3-manifolds at the root $q=e^\frac{2\pi i}{r}$. As an application, we prove the volume conjecture for large families of oriented Seifert fibered 3-manifolds with empty or non-empty boundary.

Project 2: Colaborated work on Seifert cobordisms and the Chen-Yang volume conjecture (Renaud Detcherry, Efstratia Kalfagianni, Shashini Marasinghe)

We study the large $r$ asymptotic behavior of the Turaev-Viro invariants $TV_r(M;e^2πi?r)$ of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of $∂M$. We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.

Future Directions: Topics of Interest