Abstract: There have been a lot of marvelous results revealed by string theory, which deeply relate different aspects of mathematics. All these mysterious relations are connected by a core idea in string theory called "duality". In this talk, I will focus on a duality between Chern-Simons gauge theory and topological string theory. Based on large N Chern-Simons/topological string duality, in a series of papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants in the topological string theory. I will discuss a proof of this conjecture and its application and relation to other problems, for example, the famous volume conjecture.