Seminars & Colloquia Calendar

Abstract: One of the central problems of geometry is to classify manifolds which admit special geometric structures. Calabi-Yau manifolds provide a particularly interesting class of examples. Since their discovery forty years ago they have stimulated extraordinary research activity, leading to new, beautiful mathematics and surprising connections with physics. A distinctive feature of Calabi-Yau manifolds is that they lie at the intersection of three branches of geometry: algebraic, differential, and symplectic. As a result, there is an abundance of examples of these manifolds as well as tools we use to understand them. The talk will focus on invariants of Calabi-Yau manifolds, which are defined by counting holomorphic curves. In particular, I will discuss a recent proof, joint with E. Ionel and T. Walpuski, of the Gopakumar-Vafa finiteness conjecture, which concerns the algebraic structure of these invariants. I will then outline a proposal for defining a new invariant, which counts holomorphic curves together with solutions to partial differential equations originating from gauge theory; the technical challenges of this proposal will lead us to explore uncharted territories in geometry and analysis.