PATCH Day February 17 – Etnyre and Ghrist

PATCH–the Philadelphia Area Topology, Contact and Hyperbolic Seminar, a joint venture with Bryn Mawr College, Haverford College, and now the University of Pennsylvania has its first meeting of Spring 2011 on Feb. 17.  There will be two talks in room 3C8 of Penn’s David Rittenhouse Lab starting at 4:00.

The speakers are John Etnyre of Georgia Tech and Rob Ghrist from UPenn.

Etnyre will speak on

Title: The Contact Sphere Theorem and Tightness in Contact Metric Manifolds

Abstract: We establish an analog of the sphere theorem in the setting of contact geometry. Specifically, if a given three dimensional contact manifold admits a compatible Riemannian metric of positive 4/9-pinched curvature then the underlying contact structure is tight. The proof is a blend of topological and geometric techniques. A necessary technical result is a lower bound for the radius of a tight ball in a contact 3-manifold. We will also discuss geometric conditions in dimension three for a contact structure to be universally tight in the nonpositive curvature setting. This is joint work with Rafal Komendarczyk and Patrick Massot.

Ghrist will speak on

Title: Braid Floer Homology

Abstract: The classical Arnol´d Conjecture concerns the number of 1-periodic orbits of 1-periodic Hamiltonian dynamics on a symplectic manifold.The resolution of this conjecture was the impetus for and first triumphof Floer homology. The present talk considers the problem of periodicorbits of higher periods. In the case (trivial for the Arnol´d Conjecture)of a 2-dimensional disc, these orbits are braids.

This talk describes a relative Floer homology that is a topologicalinvariant of (pairs of) braids. This can be used as a forcing theoremfor implying the existence of periodic orbits in 1-periodic Hamiltoniandynamics on a disc.

This represents joint with with J.B. van den Berg, R. Vandervorst, andW. Wojcik.

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