  In a seminal paper Richard Stanley derived Pieri rules for the Jack symmetric
function basis. These rules were extended by Macdonald to his now famous
symmetric function basis. The original form of these rules had a forbidding
complexity that made them difficult to use in explicit calculations. In the
early 90's it was discovered that, due to massive cancellations, the dual rule,
which expresses skewing by $e_1$ the modified Macdonald polynomial ${\tilde
H}_\mu[X;q,t]$, can be given a very simple combinatorial form in terms of
corner weights of the Ferrers' diagram of $\mu$. A similar formula was later
obtained by the last named author for the multiplication of ${\tilde
H}_\mu[X;q,t]$ by $e_1$, but never published. In the years that followed we
have seen some truly remarkable uses of these two Pieri rules in establishing
highly non trivial combinatorial results in the Theory of Macdonald
polynomials. This theory has recently been spectacularly enriched by various
Algebraic Geometrical results in the works of Hikita, Schiffmann,
Schiffmann-Vasserot, A. Negut and Gorsky-Negut. This development opens up the
challenging task of deriving their results by purely Algebraic Combinatorial
methods. In this paper we present the progress obtained by means of Pieri
rules.
