Berlin-Hamburg-Seminar am 22.5.2017

Doris Hein (Freiburg)  Local invariant Morse theory and applications in Hamiltonian dynamics

Local homology is a useful tool to study periodic orbits. For example, the key to the existence of infinitely many periodic orbits of Hamiltonian systems are properties of the local Floer homology of one special orbit. I will discuss a discrete version of this invariant constructed using local invariant Morse homology of a discrete action function. The construction is very geometric and relies on a hands-on description of invariant local Morse homology. The resulting local homology can be interpreted as an invariant of germs of Hamiltonian systems or of closed Reeb orbits. It has properties similar to those of local Floer homology in the symplectic setting and probably similar applications in dynamics.

Jo Nelson (Columbia University, New York)  An integral lift of contact homology

I will discuss joint work with Hutchings which gives a rigorous construction of cylindrical contact homology via geometric methods. This talk will highlight our use of non-equivariant constructions, automatic transversality, and obstruction bundle gluing. Together these yield a nonequivariant homological contact invariant which is expected to be isomorphic to SH⁺ under suitable assumptions. By making use of family Floer theory we obtain an S¹-equivariant theory defined with coefficients in Z, which when tensored with Q recovers the classical cylindrical contact homology, now with the guarantee of well-definedness and invariance. This integral lift of contact homology also contains interesting torsion information.