Mathematics Colloquia and Seminars

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First, I will define the notions of symplectic and contact structures, and show examples of manifolds which can be given one of these structures (no manifold has both). There is a duality between the two structures (geometries) in a sense that given a statement in the symplectic theory, one can get its analog in the contact case and vice versa. The situation here is similar to the duality between affine and projective geometies. Then I will be talking about lagrangian and legandrian submanifolds of symplectic and contact manifolds respectively, and singularities of their images under the corresponding projections onto the base. The concrete example is the theory of legandrian knots in 3-space. If time permits, I will mention the physical origins: configuration spaces (cotangent bundles), wave fronts etc.