The icosahedron, the Rogers-Ramanujan identities, and beyond

The first part of this lecture series is meant to illustrate the dictum that all suÿciently beautiful mathematical objects are connected. The two objects we choose to illustrate this are the icosahedron, the most subtle of the Platonic solids, and the Rogers-Ramanujan identities, often considered the most beautiful formulas in all of mathematics. These two objects are intimately related to each other and to a wide variety of topics ranging from pure number theory and the theory of modular forms to combinatorics and continued fractions to conformal field theory and mirror symmetry, with guest appearances by various other gems of mathematics like Apéry’s proof of the irrationality of (2). I will describe some of these connections and use this as the starting point for the second part of the course, whose theme is the identification of special algebraic varieties as modular varieties. The most familiar example here is of course the Taniyama-Weil conjecture as finally proved much later by Wiles et al, according to which every elliptic curve y2 = x3 +Ax+B with A and B rational can be parametrized by modular functions, but there are also beautiful higher-dimensional examples where special algebraic varieties of interest in algebraic geometry or mathematical physics are identified with Kuga-Sato, Hilbert, or Picard modular varieties, whose definitions will be explained in the course. The central example will be the “conifold fiber of the mirror quintic family,” for which various at least partial modular descriptions will be given by combining ideas from the first part of the course.