In the late 1980, in his seminal work, E. Witten introduced new invariants of 3-manifolds, by quantizing Chern-Simons quantum field theory and proposed a (formal) 3-dimensional interpretation of Jones polynomial. His achievement was to fit these invariants into a larger structure, that of a 2+1 Topological Quantum Field Theory. Based on combinatorial and topological methods the first mathematically rigorous construction of these TQFT was given by Reshetikhin-Turaev using quantum groups and alternatively later by Blanchet-Habbager-Masbaum-Vogel using Kauffman bracket and HOMFLY polynomials. The Witten-Reshetikhin-Turaev TQFT provides among other things a family of representations of the mapping class group of a surface and these representations can also be rigorously constructed via Chern-Simons gauge theory or conformal field theory using geometric methods.

Semiclassical study of WRT TQFT-invariants is an important research topic in quantum topology and it has been motivated by a fundamental problem: the relation between quantum invariants and geometry and topology of 3-manifolds. The microlocal analysis techniques for geometric quantization of Kähler manifolds and the theory of Toeplitz operators provide powerful tools to study these properties by juggling between combinatorial and geometric approaches. Asymptotic faithfulness of the quantum representations of the mapping class groups, semiclassical limit of the curve operators, geometric reformulation of AJ conjecture by studying microsupport of knot states and proving Witten conjecture on asymptotics of quantum invariants for mapping tori and for an infinite family of hyperbolic manifold obtained by surgery on the figure eight knot are some examples of the results obtained by these methods.

In a series of 4 lectures we will try to give an introduction to this subject and also present some ongoing projects and questions.