Abstract:
The course is about an interesting algebraic structure which has recently been discovered, and keeps being rediscovered in different fields of mathematics. In knot theory this structure is related to the skein algebra of the torus and produces knot invariants, as in a recent work of Morton and Samuelson. In algebraic geometry and representation theory it is related to the elliptic Hall algebra of Burban and Schiffmann, so it describes the category of vector bundles on an elliptic curve. Kontsevich and Soibelman use a similar structure to study DT invariants and wall crossing phenomena. Closely related structures are the shuffle algebra of Feigin and Tsymbaliuk and the double affine Hecke algebra (DAHA) of Cherednik. In the conjectures of Hausel, Letellier and Rodiguez-Villegas a similar structure describes the mixed Hodge structures of character varieties and moduli spaces of Higgs bundles. Finally, in algebraic combinatorics the same kind of structure produces identities between symmetric functions and enumerates Dyck paths and parking functions, as in the work of
Carlsson and myself.
This subject is very new and is developing very fast, even the main ingredient, the Macdonald polynomials were discovered quite recently (1988).
During the course all of these things will be discussed in detail. I will assume almost no preliminaries except some basic mathematical notions. It may happen that we'll have guest lectures by experts about
some of the topics above.