In 1928, Pólya proved the following remarkable theorem: Suppose p(x₁,...,x_n) is a homogeneous polynomial and suppose p(a) > 0 on the unit simplex {a_j ³ 0:Sa_j = 1}. Then for sufficiently large N, all the coefficients of (Sx_j)^Np are positive. Recently, a mathematician used a published estimate for N (in terms of the data from p) in some joint work. He later discovered that the proof of this estimate had unfixable gaps. Facing a rewrite in galleys, he went into his office in a desperate attempt to find a new proof of that estimate, or a better one, and was never seen again. All that was left were the notes that are the basis of this seminar.