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Arnold Conjectures and introduction to the generating functions technique.

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Speaker: 
Milena Pabiniak
Affiliation: 
Köln
Date: 
Thu, 2015-11-12 16:30 - 17:30
Location: 
MPIM Lecture Hall

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.

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