Berlin-Hamburg-Seminar am 15.01.2018
Oleg Lazarev (Columbia) Simplifying Weinstein Morse functions
Cieliebak and Eliashberg showed that any Weinstein Morse function on Euclidean space that is not symplectomorphic to the standard symplectic structure necessarily has at least three critical points; an infinite collection of such exotic examples were constructed by McLean. I will explain how to use handle-slides and loose Legendrians to show that this lower bound on the number of critical points is exact; that is, any Weinstein structure on Euclidean space R2n has a compatible Weinstein Morse function with at most three critical points (of index 0, n-1, and n). Furthermore, the number of gradient trajectories between the index n-1, n critical points can be uniformly bounded independent of the Weinstein structure. Similarly, any Weinstein structure on the cotangent bundle of the sphere of dimension at least three has a compatible Weinstein Morse function with two critical points. As applications, I will give new proofs of some existing h-principles and present some new constructions of exotic cotangent bundles.
Marc Kegel (Köln) The knot complement problem for Legendrian and transverse knots
The famous knot complement theorem by Gordon and Luecke states that two knots in the 3-sphere are equivalent if and only if their complements are homeomorphic.
In this talk I want to discuss the same question for Legendrian and transverse knots and links in contact 3-manifolds. The main results are that Legendrian as well as transverse knots in the tight contact 3-sphere are equivalent if and only if their exteriors are contactomorphic.