Abstract:
On page 9 of Ramanujan's first letter to Hardy in 1913, he wrote down some remarkable evaluations (for specific values of tau) of the continued fraction r(tau)=q^(1/5)/(1+q/(1+q^2/(1+q^3/(1+...))) where q=exp(2π i tau) for tau in the upper half plane. These formulas made a deep impression on Hardy, who later wrote that "I had never seen anything in the least like them before."
There is actually a wonderful geometric explanation for these evaluations, arising from the fact that the Rogers-Ramanujan continued fraction exhibits icosahedral symmetry. This symmetry, not widely known, was formally established by Duke in 2005. In this talk, I will reformulate this result in a somewhat more conceptual way. The key idea is to explicitly pin down, in a natural way, the exceptional isomorphism between the rotational symmetry group of the icosahedron and PSL(2,5).