Representations and Lie Theory SeminarAutumn 2019
Time: Wednesday, 16:15  17:15


TIME  SPEAKER  TITLE 
September 25  O. Costin  Stokes phenomenon for the beginners 
October 2  O. Costin  Stokes phenomenon: applications 
November 6  Marty Golubitsky  Symmetrybreaking and Pattern Formation 
November 13  Joshua Wen  Wreath Macdonald polynomials as eigenstates 
November 20  Matt Rupert  Unrolled quantum groups and vertex operator algebras 
November 27  No Seminar  Thanksgiving 
November 6 Pattern formation is often studied mathematically using equivariant bifurcation theory (or spontaneous symmetrybreaking). Dynamically, patterns are either stationary or timeperiodic (spatiotemporal). In this survey lecture I will describe some patterns that occur in fluids, physics, and biology and relate the patterns to equivariant bifurcation theory. The relevant bifurcation theory leads to menus of expected pattern types  each menu is driven by an irreducible representation (over the reals) of an appropriate symmetry group.
November 13 Wreath Macdonald polynomials were defined by Haiman as generalizations of transformed Macdonald polynomials from the symmetric groups to their wreath products with cyclic groups of order m. In a sense, their definition was given in the hope that they would correspond to Ktheoretic fixed point classes of cyclic quiver varieties, much like how Haiman's proof of Macdonald positivity assigns Macdonald polynomials to fixed points of Hilbert schemes of points on the plane. This hope was realized by Bezrukavnikov and Finkelberg, and the subject has been relatively untouched until now. I will present a first result exploring possible ties to integrable systems. Using work of Frenkel, Jing, and Wang, we can situate the wreath Macdonald polynomials in the vertex representation of the quantum toroidal algebra of sl_m. I will present the result that, in this setting, the wreath Macdonald polynomials diagonalize the horizontal Heisenberg subalgebra of the quantum toroidal algebraa first step towards developing a notion of 'wreath Macdonald operators'.
November 20 It was shown by Kazhdan and Lusztig in the 1990s that there exists a braided equivalence between module categories of affine Lie algebras and corresponding quantum groups. These module categories for affine Lie algebras were later realized as module categories over certain rational vertex operator algebras. In this talk I will discuss a correspondence between modules categories of the unrolled restricted quantum group of sl2 at even root of unity and the singlet vertex operator algebra, and how to use this correspondence to construct braided tensor categories related to the Bp vertex operator algebras.
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