Representations and Lie Theory Seminar
Time: Wednesday, 16:15 - 17:15
|September 25||O. Costin||Stokes phenomenon for the beginners|
|October 2||O. Costin||Stokes phenomenon: applications|
|November 6||Marty Golubitsky||Symmetry-breaking 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 symmetry-breaking). Dynamically, patterns are either stationary or time-periodic (spatio-temporal). 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 K-theoretic 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 algebra---a 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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