Abstract:
In contact topology, an important problem is to understand Legendrian submanifolds; these submanifolds are always tangent to the plane field given by the contact structure. Contact manifolds are odd dimensional manifolds with a maximally non-integrable hyperplane structure, they arise naturally from the study of constrained dynamics. For contact 3-manifolds, the contact structure is field of planes and Legendrians are 1 dimensional submanifolds, while for contact 5-manifolds, the contact structure is a field of 4-dimensional planes and Legendrians are 2-dimensional submanifolds. I will give a brief overview of Legendrian links and surfaces: how to draw them, and Legendrian invariants (arising from Floer theory). I will also talk about how Legendrian invariants are used to distinguish and obstruct exact Lagrangian surfaces in the symplectic 4-ball.