Mathematics Colloquia and Seminars

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Let C be a smooth algebraic curve defined over the complex numbers. The theory developed by Eynard, and Orantin, that has been largely influenced by topological string theory due to Marino and his collaborators, constructs a B-model topological string theory from the data (C, x, y). Here x and y are meromorphic functions on C. The surprise is that the mirror symmetric dual to this theory is predicted to give all open and closed Gromov-Witten invariants of arbitrary toric Calabi-Yau 3-folds. Physics speculations go beyond. If x and y satisfy that the Steinberg symbol {x, y} in the second K-group of the function field of C is a torsion, then a variant of the theory due to Borot and Eynard (after Fuji, Gukov and Sulkowski) should give the colored HOMFLY polynomials of arbitrary knots (after taking a non-perturbative quantization using Riemann theta functions). The talk surveys some of the physics conjectures. I will also explain the relation between the Bloch regulator and the Eynard-Orantin differentials. So far mathematical theorems are established only when C has genus 0.