Mathematics Colloquia and Seminars

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The purpose of this talk is to relate the geometries of calibrated submanifolds to their gauge theories. We study the moduli space of deformations of a special kind of associative submanifolds in a $G_2$ manifold (which we call complex associative submanifolds); and we study the moduli space of deformations of a special kind of Cayley submanifolds (which we call complex Cayley submanifolds). We show that deformation spaces can be perturbed to be smooth and finite dimensional. We also get similar results for the deformation spaces of other calibrated submanifolds. We discuss the relation to Seiberg-Witten theory and other generalizations. We propose a certain counting invariant for associative and Cayley submanifolds of foliated manifolds.