# University of Toronto's Symplectic Geometry Seminar

Monday February 28, 2005, 14:10--15:00

SS 5017A

## Martin Pinsonnault

###
Fields Institute

##
Maximal tori in the group of Hamiltonian automorphisms
of 4-manifolds

** Abstract: **
Maximal tori are central in the theory of compact Lie
groups. A
basic theorem asserts that all these tori are conjugate. For
diffeomorphism groups, this theorem is cannot hold since conjugacy
classes
of maximal tori correspond to geometrically distinct torus actions, and
compact manifolds such as S^2 x S^2 admits infinitely many
maximal
T^2-actions. However, we will see that on a symplectic 4-manifold,
there are only finitely many conjugacy classes of maximal tori in the
infinite dimensional group of Hamiltonian diffeomorphisms. This result
gives an other indication that symplectomorphism groups are, to some
extend, infinite dimensional analogs of compact Lie groups. The proof
uses
two main ingredients: Delzant's theory of toric manifolds and Gromov's
theory of J-holomorphic curves.