Diffeomorphisms in symplectic category posses certain rigidity properties
(symplectic camel, Gromov non-squeezing theorem). An important
manifestation of rigidity is given by the conjectures posed by V. Arnold
describing a lower bound for the number of fixed points of a Hamiltonian
diffeomorphism h (i.e. symplectic diffeomorphism Hamiltonian isotopic to
identity) of a compact symplectic manifold. These lower bounds
are greater than what topological arguments could predict.
Arnold Conjectures present a difficult problem and motivated a lot of important
research in symplectic geometry. The case of C¹-small Hamiltonian diffeomorphisms
is easy to prove. Generalization of these ideas gave rise to the technique of generating
functions.
In this talk I will introduce generating functions and sketch a proof of
Arnold Conjecture for a complex projective space.
Then I will formulate a version of Arnold Conjecture in contact geometry
setting describing the minimal number of translated points of a contact
isotopy (posted by S. Sandon) and describe a current work in progress, with
G. Granja, Y. Karshon and S. Sandon, aimed to prove this conjecture for
lens spaces and other contact toric manifolds.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/3050