Counting holomorphic curves in algebraic varieties is an ancient subject of algebraic geometry under the name of enumerative algebraic geometry. Thirty years ago, it appeared in physics as a key physical invariant in so called topological string. Since then, there has been a tremendous amount of effort to compute them by both mathematician and physicists. The genus zero computation in 90's leaded to the birth of mirror symmetry as a mathematical subject. However, the computation in higher genus turns out to be one of most difficult problems in geometry and physics. For the quintic 3-fold (simplest example of Calabi-Yau 3-fold), our knowledge stops at genus one despite of many hard work during last twenty years. Nevertheless, physicist have proposed a zoo of incredible conjectures for both the structure and explicit formula. Working with a group of talented young mathematician, we made a breakthrough on the problem recently. I will give an overview of the
program in the talk. Speaker(s): Yongbin Ruan (University of Michigan)