I prove a Torelli theorem for certain Laurent polynomials. This provides strong evidence for the idea that, under mirror symmetry, a Fano manifold corresponds to a single geometric object called a cluster variety. As things stand, mirror symmetry provides a one-to-many correspondence between a single Fano manifold and a collection of Laurent polynomials (or Landau–Ginzburg models); my result gives a geometric proof that, for smooth Fanos in dimension two, these Laurent polynomials assemble to give a single cluster variety.

My other theorem is joint work with Tom Coates and Qaasim Shafi, and determines, under mild hypotheses, how the genus-zero Gromov–Witten invariants of a space X change under blow-ups of X. This is a significant result in enumerative geometry; it also expands the range of Fano manifolds for which we can establish mirror symmetry.