In 1991, Birch outlined a method for computing linear operators on the free module defined on the isometry classes of a positive definite ternary quadratic form. Through a series of computations and observations, he conjectured that this module, equipped with infinitely many commuting linear operators, is isomorphic to a subspace of classical modular forms. In this talk, we describe the mathematical background necessary to understand Birch's conjecture, and then outline a proof in the special case of squarefree discriminant.