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Nikita Nekrasov : Four-Manifolds, Symplectic Geometry, and Mirror Symmetry

Some of the old problems in algebraic geometry, as well as relatively new problems in the theory of quantization, were solved using topological sigma models. The sigma models describe maps of a manifold M to a target space X. It is very well-known that no sensible theory exists when the dimension of M is greater than two. In my talk I will try to argue in favor of the existence of an interesting theory of maps in the case where M is a four-dimensional Riemannian manifold and X is a classifying space of some compact Lie group (or its finite-dimensional model). To get there we will need to introduce & develop certain aspects of Donaldson theory and higher-dimensional analogues of Whitman hierarchies. No knowledge of Donaldson theory or Whitman hierarchies is necessary.

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