The refined Chern-Simons theory is a one-parameter deformation of the ordinary Chern-Simons theory on Seifert manifolds. It is defined via an index of the theory on N M5 branes, where the corresponding one-parameter deformation is a natural deformation of the geometric background. Analogously with the unrefined case, the solution of refined Chern-Simons theory is given in terms of S and T matrices, which are the proper Macdonald deformations of the usual ones. This provides a direct way to compute refined Chern-Simons invariants of a wide class of three-manifolds and knots. The knot invariants of refined Chern-Simons theory are conjectured to coincide with the knot superpolynomials -Poincare polynomials of the triply graded knot homology theory. This conjecture is checked for a large number of torus knots in S 3 , colored by the fundamental representation. This is a short, expository version of arXiv:1105.5117, with some new results included. 1 1 Based on talks presented by M.A. at several conferences and workshops, including String-Math 2011 Conference at U. of Pennsilvania.

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Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q, t ∈ [0, 1). We prove several results about these processes, which include the following.(1) We explicitly evaluate expectations of a rich family of observables for these processes. (2) In the case t = 0, we find a Fredholm determinant formula for a q-Laplace transform of the distribution of the last part of the Macdonald-random partition. (3) We introduce Markov dynamics that preserve the class of Macdonald processes and lead to new "integrable" 2d and 1d interacting particle systems. (4) In a large time limit transition, and as q goes to 1, the particles of these systems crystallize on a lattice, and fluctuations around the lattice converge to O'Connell's Whittaker process that describe semi-discrete Brownian directed polymers.

AGT correspondence relates a class of 4d gauge theories in four dimensions to conformal blocks of Liouville CFT. There is a simple proof of the correspondence when the conformal blocks admit a free field representation. In those cases, vortex defects of the gauge theory play a crucial role, extending the correspondence to a triality. This makes use of a duality between 4d gauge theories in a certain background, and the theories on their vortices. The gauge/vortex duality is a physical realization of large N duality of topological string which was conjectured in [1] to provide an explanation for AGT correspondence. This paper is a review of [2], written for the special volume edited by J. Teschner.