Course on knots, Hall algebras and symmetric functions

[Anton Mellit]

Dear All,

On Fridays from April 15 till May 20,
at 14:00-16:00, Room 134 of SISSA

I will teach a course "Knots, Hall algebras and symmetric functions".

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.

The first lecture is at 14:00 on Friday, April 15 in the room 134 (SISSA)

Best regards,
Anton Mellit