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SUMMARY:Hofer geometry at large - Adrian Dawid\, University of Cambridge
DTSTART:20240517T150000Z
DTEND:20240517T160000Z
UID:TALK213367@talks.cam.ac.uk
CONTACT:Alexis Marchand
DESCRIPTION:On a symplectic manifold M\, the group of Hamiltonian diffeomo
 rphisms Ham(M) is a natural object of study. In classical mechanics\, when
  M models the phase space of a system\, Ham(M) arises as the group describ
 ing all admissible motions. In 1990\, Hofer introduced a bi-invariant metr
 ic on this group. Intuitively\, the Hofer distance between a Hamiltonian d
 iffeomorphism and the identity can be thought of as the minimal energy nec
 essary to generate the diffeomorphism. One can also extend this notion to 
 a metric between Lagrangian submanifolds in the same Hamiltonian isotopy c
 lass. The Hofer distance between them should be thought of as the minimal 
 amount of energy that has to spent to deform one into the other. The metri
 c properties of these spaces remain largely elusive. However\, they are ve
 ry sensitive to the symplectic topology of the ambient space and the Lagra
 ngian. This can often be seen through the large-scale behaviour of the Hof
 er metric. Here one usually looks at the space of all Lagrangians in a spe
 cific Hamiltonian isotopy class.  In some situations\, the diameter of thi
 s space endowed with the Hofer metric is finite. In other situations\, it 
 admits quasi-isometric embeddings of infinite dimensional normed vector sp
 aces. We will outline several examples of these large-scale phenomena. If 
 time permits\, we will discuss the construction of such quasi-isometric em
 beddings for a specific example.
LOCATION:MR13
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