The shuffle conjecture gives a combinatorial interpretation of certain generating functions arising from the study of the action of the permutation group S_n on the algebra of polynomials in 2n variables x_1, y_1,..., x_n, y_n. The combinatorial side is given in terms of certain objects called parking functions, and the generating functions that arise there can be thought of as generalizations of Catalan numbers. The conjecture was formulated by Haglund and his collaborators in 2005, and since then attracted a lot of interest of algebraic combinatorists. During Erik Carlsson's stay at ICTP last year we were able to prove the conjecture, and that's what the talk will be about. First 50 minutes are intended for a general audience, I will talk about symmetric functions and demonstrate some important general tools for dealing with the generating functions of the kind we encountered. After a short break I will continue with more details on our proof.
Schedule:
16:00 - 16:50; Part I - Basic introduction
16:50 - 17:00; Coffee break
17:00 - 17:50; Part II - Technical discussion