# University of Toronto's Symplectic Geometry Seminar

Oct. 20 2003, 2:10 - 3

SS5017A

## Anne-Laure Biolley

###
University of Toronto

##
"The interplay between Floer cohomology, symplectic hyperbolicty and
(almost)-complex hyperbolicity"

** Abstract: **
Symplectic manifolds can be naturally provided with compatible
almost-complex structures, which allow one to define
pseudo-holomorphic curves. On the one hand, they are a powerful tool in
probing symplectic geometry and one of the ingredients for defining Floer
cohomology. On the other hand, on the complex side, they lead to the
definition of (almost)-complex hyperbolicity. Going further, Floer
cohomology allows one to define a notion of symplectic hyperbolicity. This
naturally raises the issue of possible links between the symplectic
hyperbolicity of a symplectic manifold and the complex hyperbolicity of
its compatible almost complex structures. I will present the interplay
between these two notions. Then, I'll explain how this analysis allows one to
get stability results for non complex hyperbolicity under deformation of
the almost complex structure among the set of structures compatible to a
fixed symplectic structure, thus getting a generalisation of Bangert's
result which addressed the particular case of the standard torus.