Mirror symmetry, affine geometry and Gromov-Hausdorff limits

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Mirror Symmetry is a phenomenon discovered by physicists around 1990. It relates different geometries (symplectic and holomorphic) on pairs of different spaces. This proposal concerns differential-geometric aspects of a joint program with Mark Gross (UCSD) for a comprehensive explanation of this phenomenon by a notion of dual degeneration limits. The program also points to a dictionary of independent interest between symplectic and holomorphic geometry, respectively, on one side and integral affine geometry on the other side.