2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/66466This article accompanies my ICM talk in August 2002. Three conjectural directions in Gromov-Witten theory are discussed: Gorenstein properties, BPS states, and Virasoro constraints. Each points to basic structures in the subject which are not yet understood.10 pagesAlgebraic Geometrytext