Georgia Institute of Technology College of Sciences

John Etnyre: "Embeddings of contact manifolds" Abstract: I will discuss recent results concerning embeddings and isotopies of one contact manifold into another. Such embeddings should be thought of as generalizations of transverse knots in 3-dimensional contact manifolds (where they have been instrumental in the development of our understanding of contact geometry). I will mainly focus on embeddings of contact 3-manifolds into contact 5-manifolds. In this talk I will discuss joint work with Ryo Furukawa aimed at using braiding techniques to study contact embeddings. Braided embeddings give an explicit way to represent some (maybe all) smooth embeddings and should be useful in computing various invariants. If time permits I will also discuss other methods for embedding and constructions one may perform on contact submanifolds. Dan Cristofaro-Gardiner: "Beyond the Weinstein conjecture" Abstract: The Weinstein conjecture states that any Reeb vector field on a closed manifold has at least one closed orbit. The three-dimensional case of this conjecture was proved by Taubes in 2007, and Hutchings and I later showed that in this case there are always at least 2 orbits. While examples exist with exactly two orbits, one expects that this lower bound can be significantly improved with additional assumptions. For example, a theorem of Hofer, Wysocki, and Zehnder states that a generic nondegenerate Reeb vector field associated to the standard contact structure on $S^3$ has either 2, or infinitely many, closed orbits. We prove that any nondegenerate Reeb vector field has 2 or infinitely many closed orbits as long as the associated contact structure has torsion first Chern class. This is joint work with Mike Hutchings and Dan Pomerleano.

The point-pushing subgroup of the mapping class group of a surface with a marked point can be considered topologically as the subgroup that pushes the marked point about loops in the surface. Birman demonstrated that this subgroup is abstractly isomorphic to the fundamental group of the surface, \pi_1(S). We can characterize this point-pushing subgroup algebraically as the only normal subgroup inside of the mapping class group isomorphic to \pi_1(S). This uniqueness allows us to recover a description of the outer automorphism group of the mapping class group.

Dehn surgery is a fundamental tool for constructing oriented 3-Manifolds. If we fix a knot K in an oriented 3-manifold Y and do surgeries with distinct slopes r and s, we can ask under which conditions the resulting oriented manifold Y(r) and Y(s) might be orientation preserving homeomorphic. The cosmetic surgery conjecture state that if the knot exterior is boundary irreducible then this can't happen. My talk will be about the case where Y is an homology sphere and K is an hyperbolic knot.

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal R matrix, and we obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant is an invariant of framed links, and is constructed diagrammatically using a ribbon Hopf algebra. In that construction, a copy of the universal R matrix is attached to each positive crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal R matrix. On the other hand, R. Kashaev showed that the Heisenberg double has the canonical element (the universal S matrix) satisfying the pentagon relation. In this talk we reconstruct the universal quantum invariant using Heisenberg double, and extend it to an invariant of colored ideal triangulations of the complement. In this construction, a copy of the universal S matrix is attached to each tetrahedron and the invariance under the colored Pachner (2,3) move is shown by the pentagon equation of the universal S matrix