Fock space and block space: using vertex operators to prove MacMahon's plane partition generating function

In the early 1900's MacMahon proved a beautiful product formula for the generating function for plane partitions. I'll be presenting my favorite proof of this theorem (due to Okounkov and others), which uses vertex operator formalism. I promise, it's more fun than it sounds!
This proof technique has many other applications. Time permitting, I hope to discuss a couple of these tomorrow, including a proof of the Cauchy identity for Schur functions and a nice product formula for the generating function for plane partitions in a room with an infinite baseboard.