  Macdonald processes are probability measures on sequences of partitions
defined in terms of nonnegative specializations of the Macdonald symmetric
functions and two Macdonald parameters q,t in [0,1). We prove several results
about these processes, which include the following.
  (1) We explicitly evaluate expectations of a rich family of observables for
these processes. (2) In the case t=0, we find a Fredholm determinant formula
for a q-Laplace transform of the distribution of the last part of the
Macdonald-random partition. (3) We introduce Markov dynamics that preserve the
class of Macdonald processes and lead to new "integrable" 2d and 1d interacting
particle systems. (4) In a large time limit transition, and as q goes to 1, the
particles of these systems crystallize on a lattice, and fluctuations around
the lattice converge to O'Connell's Whittaker process that describe
semi-discrete Brownian directed polymers. (5) This yields a Fredholm
determinant for the Laplace transform of the polymer partition function, and
taking its asymptotics we prove KPZ universality for the polymer (free energy
fluctuation exponent 1/3 and Tracy-Widom GUE limit law). (6) Under intermediate
disorder scaling, we recover the Laplace transform of the solution of the KPZ
equation with narrow wedge initial data. (7) We provide contour integral
formulas for a wide array of polymer moments. (8) This results in a new ansatz
for solving quantum many body systems such as the delta Bose gas.
