# Mathematics Colloquia and Seminars

I'll explain recent joint work with Jim Haglund and Nick Loehr, in which we prove a a combinatorial formula for the Macdonald polynomial $\tilde{H}_{\mu }(x;q,t)$ which had been conjectured by Haglund. Such a combinatorial formula had been sought ever since Macdonald introduced his polynomials in 1988.
In general, our formula doesn't yet give a new proof of the positivity theorem for Macdonald polynomials, because it expresses them in terms of monomials, rather than Schur functions. However, it does yield a new combinatorial expression for the Schur function expansion when the partition $\mu$ has parts $\leq 2$, and there is hope to extend this result.