Room P3.10, Mathematics Building

Nick Sheridan, Princeton
Counting curves using the Fukaya category

In 1991, string theorists Candelas, de la Ossa, Green and Parkes made a startling prediction for the number of curves in each degree on a generic quintic threefold, in terms of periods of a holomorphic volume form on a “mirror manifold”. Givental and Lian, Liu and Yau gave a mathematical proof of this version of mirror symmetry for the quintic threefold (and many more examples) in 1996. In the meantime (1994), Kontsevich had introduced his “homological mirror symmetry” conjecture and stated that it would “unveil the mystery of mirror symmetry”. I will explain how to prove that the number of curves on the quintic threefold matches up with the periods of the mirror via homological mirror symmetry. I will also attempt to explain in what sense this is “less mysterious” than the previous proof.