Chern-Simons theory: duality, integrability and topology

I will talk about what interesting directions there are for research related to the Chern-Simons theory. First, Chern-Simons theory is one of the simplest three-dimensional quantum field theories that we can solve exactly using quantum algebra and the R-matrix. The question of searching for integrable structures in other gauge theories seems promising. Second, vacuum expectation values in Chern-Simons theory are invariants of knots (or links) in a given 3-dimensional manifold. Since the theory of knots and their invariants is far from its complete description, this question is of great interest. Third, there are two dualities with the Chern-Simons theory. One holographic duality between 3D Chern-Simons theory and 2D conformal WZW theory. The second duality arises from string theory and is a mirror symmetry between the A-model (Chern-Simons theory) and the B-model (Gromov-Witten theory). Due to the progress in the Chern-Simons theory, it is possible to investigate this duality in explicit terms.