Georgia Institute of Technology College of Sciences

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

In this series of talks I will begin by discussing the idea of studying smooth manifolds and their submanifolds using the symplectic (and contact) geometry of their cotangent bundles. I will then discuss Legendrian contact homology, a powerful invariant of Legendrian submanifolds of contact manifolds. After discussing the theory of contact homology, examples and useful computational techniques, I will combine this with the conormal discussion to define Knot Contact Homology and discuss its many wonders properties and conjectures concerning its connection to other invariants of knots in S^3.

A well-known problem in discerte convex geometry, attributed to the Dutch painter Durrer and first formulated by G. C. Shephard, is concerned with whether every convex polyope P in Euclidean 3-space has a simpe net, i.e., whether the surface of P can be isometrically embedded in the Euclidean plane after it has been cut along some spanning tree of its edges. In this talk we show that the answer is yes after an affine transformation. In particular the combinatorial structure of P plays no role in deciding its unfoldability, which settles a question of Croft, Falconer, and Guy. The proof employs a topological lemma which provides a criterion for checking embeddedness of immersed disks.