Cherednik Algebras and Applications Learning Seminar Spring 2021

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Time and location information

Time: Wednesday at 5:00 - 6:15 pm
Organizer: Sam DeHority
Zoom info: The zoom link is https://columbiauniversity.zoom.us/j/98248875411. Please email me at samdehority@math.columbia.edu to be added to the mailing list and for the zoom password.

References and topics

The basic plan for the semester is to first cover the fundamentals of the structure of Cherednik algebras focusing on examples and then cover a variety of applications based on the interests of the speaker.

A rough list of topics is as follows:

  1. Review of Hecke/ affine Hecke algebras
  2. Definition of Cherednik Algebra, PBW theorem, spherical subalgebra
  3. Rational and trigonometric degenerations, Dunkl operators, relationship with integrable systems
  4. Macdonald polynomials, Macdonald constant term conjecture

For these topics general references are https://arxiv.org/abs/math/0404307 and the associated book, Double affine Hecke algebras by Cherednik, the notes https://arxiv.org/abs/1001.0432, and a recent seminar (elsewhere) with notes https://web.northeastern.edu/iloseu/DAHAEHA.html.

along with many other possibilities and applications depending on interest:

  1. Geometric rep theory and affine Springer fibers https://arxiv.org/abs/math/0207127, https://arxiv.org/abs/0705.2691, etc.
  2. Relationship with the elliptic hall algebra https://arxiv.org/abs/1202.2756 and see Northeastern/MIT notes above
  3. Knot invariants, superpolynomials
  4. etc.

List of talks

Date Talk Info
Wednesday
Feb. 10
Speaker: Álvaro Martínez
Title: Introduction to Hecke algebras and affine Hecke algebras
Abstract: We will motivate the appearance of (affine) Hecke algebras in Lie theory and discuss their basic properties and representations.
Notes: pdf
Wednesday
Feb. 17
Speaker: Henry Liu
Title: DAHAAHAHA!
Abstract: We’ll define a DAHA and its polynomial representation, which will provide some motivation for the D(oubling) and also some structural properties of the algebra.
Notes: pdf
Wednesday
Feb. 24
Speaker: Davis Lazowski
Title: Degenerate DAHA and integrable systems.
Abstract: I will discuss how the rational degeneration of DAHA is related to Olshanetsky-Perelomov Hamiltonians, Bessel functions and Knizhnik-Zamolodchikov equations.