This project aims at studying the relationship between Teichmuller theory and quantum invariants of knots and three-manifolds. The Teichmuller space, defined as the space of hyperbolic metrics on a surface up to isotopy, is a key object in understanding the geometry and ultimately the topology of low-dimensional manifolds. Its quantization, giving rise to the so-called quantum Teichmuller space, finds its origin in mathematical physics, more specifically in the study of 2+1 quantum gravity. Two approaches will be developed in understanding this connection more deeply. The first approach will consist in studying the possibility of constructing a finite dimensional modular functor associated to the representation theory of the quantum Teichmuller space. If this approach is successful, it would lead to the construction of a family of representations of the mapping class groups of punctured surfaces and ultimately to the construction of a topological quantum field theory. The second approach will consist in studying the relationship between the quantum Teichmuller space and the skein algebra of arcs and links developed recently by the PI and a collaborator. The possible connections between the two points of view is expected to be similar to the relationship between the geometrical and the combinatorial approach to SU(2)-TQFT.
The study of three-manifolds is in essence the study of the possible shapes of the universe and as such sits naturally at the intersection of mathematics and theoretical physics. The subject experienced several major breakthrough in the past thirty years following two a priori unrelated approaches: one coming from the study of the possible geometries a three-manifold admits and the other coming from mathematical physics and the notion of quantum invariants. In recent years, numerous results and conjectures have aimed at reconciling the two points of view. The PI intends on studying these connections further with the goal of translating methods from one to the other.
