It is well-known that the spectral data of the Gaudin model associated to a finite semisimple Lie algebra is encoded by the differential data of certain flat connections associated to the Langlands dual Lie algebra on the projective line with regular singularities, known as oper/Gaudin correspondence. Recently, some progress has been made in understanding the correspondence associated with affine Lie algebras. I will present a physical perspective from Kondo line defects, physically describing a local impurity chirally coupled to the bulk 2d conformal field theory. The Kondo line defects exhibit interesting integrability properties and wall-crossing behaviors, which are encoded by the generalized monodromy data of affine opers. In the physics literature, this reproduces the known ODE/IM correspondence. I will explain how the recently proposed 4d Chern Simons theory provides a new perspective which suggests the possibility of a physicists’ proof. 

 In this talk I will discuss an algorithm to piecewise dualise linear quivers into their mirror duals. This applies to the 3d N=4 version of mirror symmetry as well as its recently introduced 4d counterpart, which I will review. The algorithm uses two basic duality moves, which mimic the local S-duality of the 5-branes in the brane set-up of the 3d theories, and the properties of the S-wall. The S-wall is known to correspond to the N=4 T[SU(N)] theory in 3d and I will argue that its 4d avatar corresponds to an N=1 theory called E[USp(2N)], which flows to T[SU(N)] in a suitable 3d limit. All the basic duality moves and S-wall properties needed in the algorithm are derived in terms of some more fundamental Seiberg-like duality, which is the Intriligator--Pouliot duality in 4d and the Aharony duality in 3d.