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Gromov-Witten Theory: Mirror Symmetry, Modular Forms, and Integrable Systems

Final Report Summary - GWT (Gromov-Witten Theory: Mirror Symmetry, Modular Forms, and Integrable Systems)

The Gromov-Witten invariants of a space X record the different ways that surfaces can be arranged within X. This gives important information about the shape of X. During this project we developed powerful new techniques for computing Gromov-Witten invariants, and used these to discover and establish connections between Gromov-Witten theory and a number of other areas of mathematics. Our most striking discovery was a completely unexpected connection between Gromov-Witten theory and the classification of Fano manifolds. Fano manifolds are "atomic pieces" of mathematical shapes, and the classification of Fano manifolds is an important problem in geometry which has been open for nearly 100 years. Our work potentially opens the way to the classification of Fano manifolds in all dimensions - and thus to building a "Periodic Table of shapes".