Seminário Informal de Geometria 

14/10/2014, 11:00 — 12:30 — Sala P3.10, Pavilhão de Matemática
Milena Pabiniak, Instituto Superior Tecnico

Arnold Conjectures and introduction to the generating functions technique.

Symplectic diffeomorphisms possess certain rigidity properties (symplectic camel, 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 and a lower bound for the number of intersection points of a Lagrangian submanifold $L$ and $h(L)$. These lower bounds are greater than what topological arguments could predict.

The conjectures have been proved in many special cases, but not in full generality.

We will sketch the proof for $C^1$-small Hamiltonian diffeomorphisms. Then we introduce the technique of generating families which can be used to prove Arnold Conjectures for other Hamiltonian diffeomorphisms. If time permits we introduce contact version of Arnold Conjecture about the minimal number of translated points of a contactomorphism, and explain how to adjust the proof of Arnold Conjecture for $CP^n$ to obtain a proof of contact Arnold Conjecture for $RP^{2n-1}$.