2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67292I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of intersection numbers of the moduli spaces of flat connections on a Riemann surface by using heat kernels. (3) The mirror principle about counting curves in Calabi-Yau and general projective manifolds by using hypergeometric series.Algebraic GeometryQuantum Algebra53D30, 57R91, 81T13, 81T30text