Tuesday, October 24th, 2023
03:00pm - 04:00pm, in Wheatley 03-154-28 Mathematics Department Seminar and Common Room

Umut Varolgunes

Bogazici University, Turkey

From Classical Mechanics to Symplectic Rigidity (and Back?)

Abstract: Consider a particle moving in Euclidean space under the influence of a Hamiltonian energy function. All possible trajectories of this particle define a flow on the phase space R2 x ...x R2, where we paired each position coordinate with its corresponding momentum coordinate. One can assign to each (oriented) patch of surface in the phase space its symplectic area: add up the signed areas of the projections to each R2 factor. The birth of symplectic geometry is the observation that any Hamiltonian flow preserves these symplectic areas. A symplectic manifold is a generalization of this phase space structure to spaces with more interesting topology, e.g. on a three holed torus a symplectic structure is equivalent to an area form. I will outline some recent results (including some of mine) in symplectic geometry, restricting myself to phase spaces and surfaces.