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Abstract: This talk is intended for introducing the audience to an exciting new development in the study of Gromov-Witten theory and combinatorial topology of moduli spaces of Riemann surfaces. New results are reported in these areas using a mysterious technique called the "Eynard-Orantin Topological Recursion Formula." This idea stemmed out from a statistical mechanics work in random matrix theory. It was then applied to algebraic geometry by string theorists. But these "results" are merely conjectures because there have been no mathematical proofs for them, until very recently. In a joint work with Naizhen Zhang and others, I solved the first and very difficult conjecture among them. Our discovery is that the key mathematical technique is the Laplace transform, which plays the role of "mirror symmetry." This idea is also applicable to some combinatorial topology questions in moduli theory of Riemann surfaces. In this talk I will illustrate the whole theory by explaining the simplest example that I discovered with an undergraduate student Kevin Chapman this past summer. As you can see, the student power is amazing when we attack new problems and new conjectures!