Mathematics Colloquia and Seminars

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Contact geometry is the study of geometric structures on odd dimensional smooth manifolds given by a hyperplane field specified by a one form which satisfies a maximum nondegeneracy condition called complete non-integrability; these hyperplane fields are called contact structures. The associated one form is called a contact form and uniquely determines a vector field called the Reeb vector field on the manifold. Contact and symplectic geometry are closely intertwined and as in symplectic topology one can make use of J-holomorphic curves to obtain a topological approach, known as contact homology, to constructing contact invariants. Despite the many analytic pitfalls along the way to defining contact homology, the speaker's seminar talk will then explain how recent work of herself and Hutchings has managed to redeem this theory in dimension 3 for dynamically convex contact manifolds. Additionally this talk will have lots of cool pictures and animations illustrating these fascinating concepts in contact geometry and many concrete examples will be given.