An overview of the geometry and combinatorics of the Macdonald polynomial and q-t Catalan number

We will explore the combinatorial and geometric properties related to the Macdonald polynomials and the diagonal harmonics. We have the combinatorial Macdonald polynomial formula that fits the defining conditions directly. The shuffle conjecture gives an elegant expression of the Frobenius series of the diagonal harmonics. While the geometric properties of the Hilbert scheme and schemes over it provides explanations from a different perspective. We use examples to show that these two approaches arrive at the same goal.