The Rogers-Ramanujan identities and the icosahedron - Lecture 1

Don Zagier (Max Planck/ICTP)

The two identities

∞∑n=0xn2(1−x)· · ·(1−xn)=∏n≡±1 (mod 5)11−xn,∞∑n=0xn(n+1)(1−x)· · ·(1−xn)=∏n≡±2 (mod 5)11−xn,

discovered independently by Leonard Rogers and by Srinivasa Ramanujan more than a hundred years ago,are considered by many to be the most beautiful pair of formulas in all of mathematics. Their most strikingaspect is the unexpected appearance of the number “5”, whichalso appears in the theory of the two mostcomplicated of the Platonic solids, the icosahedron and thedodecahedron. It turns out that these twoappearances are intimately and intricately related through a wide variety of topics ranging from pure numbertheory and the theory of modular forms to combinatorics and continued fractions to conformal field theoryand mirror symmetry, with guest appearances by various other gems of mathematics like Ap ́ery’s proof of theirrationality ofζ(2). In this series of talks, which will comprise three or four lectures depending on audienceinterest and also on whether I succeed in completing a planned application to the theory of the mirror quinticof Candelaset al, I want to give a survey of some of these topics that is intended to be accessible and ofinterest to mathematicians of all levels and persuasions, without any particular prerequisites.