Intensive Lecture on Macdonald Polynomials :

                      Jim Haglund (University of Pennsylvania)

• Lecture 1. May 14, 3:30pm - 5:00pm (KIAS 1424)
  

  The Combinatorics of Macdonald Polynomials and Related Symmetric Functions

  
  Abstract : Macdonald polynomials are symmetric functions in a set of variables $X$ which depend on two extra parameters $q$, $t$. They satisfy an orthogonality relation and are closely connected to important algebraic and geometric objects such as the Hilbert scheme of points in the plane and Cherednik algebras. Macdonald's original definition was rather indirect, but in this talk we discuss a purely combinatorial formula for Macdonald polynomials due to Haiman, Loehr and the speaker. We then outline several nice applications of this formula to symmetric function theory.
  
  

• Lecture 2. May 15, 3:30pm - 5:00pm (KIAS 1424)
  

  The Combinatorics of the Space of Diagonal Harmonics

  
  Abstract : Garsia and Haiman pioneered the study of the space of Diagonal Harmonics $DH_n$ and its $S_n$ character. Using properties of the Hilbert scheme, Haiman eventually proved a wonderful formula for this character as a sum of Macdonald polynomials involving rational coefficients in $q$, $t$. Various special cases of this formula result in $q$,$t$-versions of classical combinatorial objects such as Catalan numbers. In this talk we overview results and conjectures developed over the last ten years, by a variety of researchers, involving combinatorial expressions for $q$,$t$-Catalan numbers and other objects connected to $DH_n$. In particular we discuss recent results expressing these objects in terms of constant-term identities for Laurent series.