Creutzig-Diaconescu-Ma conjectured that there exists a homomorphism from the shifted affine Yangian to the universal enveloping algebra of an iterated W-algebra. They also conjectured that this homomorphism will induce a resolution of the generalized AGT conjecture. In this talk, I will explain how to construct a homomorphism from the affine Yangian of type A to the universal enveloping algebra of a W-algebra including the non-rectangular W-algebra. I expect that this homomorphism can be extended to the shifted affine Yangian.
A universality of local eigenvalue statistics of Hermitian random matrices has been the subject of immense investigation over the past couple decades resulting in the now well established principle that local eigenvalue statistics are determined only by the symmetry class of the matrix (i.e. real symmetric vs complex Hermitian) and the shape of the local eigenvalue density. For example, generically at the edge of the spectrum the eigenvalue density of random matrices exhibits square-root decay, leading to the local eigenvalue statistics falling into the famous Tracy--Widom universality class. For non-Hermitian matrices substantially less is known, with full universality in the bulk and at the edge for a random matrix X with centered independent identically distributed entries (or IID matrix) established only quite recently. While this is quite a substantial accomplishment, it is nowhere near the generality available for Hermitian matrices. In this talk we will consider the edge statistics of a deformed IID matrix A+X, where A is some deterministic deformation of the IID matrix X. By using matrices with Gaussian entries as a middle step for comparison we will see that this model has implications for many important non-Hermitian random matrix models. This is based on joint work with Giorgio Cipolloni, Laszlo Erdos, and Hong Chang Ji.