Gromov-Witten theory: mirror symmetry, birational geometry, and the classification of fano manifolds
The classification of Fano manifolds is a long-standing and important open problem.  Fano manifolds are basic building blocks in geometry: they are `atomic pieces' of mathematical shapes.  We will take a radically new approach to Fano classification, combining Mirror Symmetry (a circle of ideas which originated in string theory) with new methods in geometry and massively-parallel computational algebra.  

Our main geometric tool will be Gromov-Witten invariants.   The Gromov-Witten invariants of a space X record the number of curves in X of a given genus and degree which meet a given collection of cycles in X; they have important applications in algebraic geometry, symplectic topology, and theoretical physics.  We will develop powerful new methods for computing Gromov-Witten invariants, and will apply these methods to Fano classification and to questions in birational geometry.