Everyone welcome. Refreshments will be served in the Math
Lounge before the exam.
Friday, June 15, 2012, 11:10 a.m., in BA 6183, 40 St. George St.
PhD Candidate: Jordan Watts
PhD Advisor: Yael Karshon
Thesis Title: Diffeologies, Differential Spaces, and Symplectic Geometry
(http://www.math.toronto.edu/jwatts/storage/ut-thesis.pdf)
Thesis Abstract:
The notion of smoothness is well-understood on manifolds, but in
practice one often requires some kind of smooth structure on subsets
or quotients of manifolds as well. Many notions of a smooth structure
have been defined for more general spaces than manifolds to remedy
this issue; for example, diffeological, differential, and Frölicher
structures are defined on arbitrary sets.
We will focus on the application of diffeology to a problem in
symplectic geometry; in particular, to differential forms on a
symplectic quotient. A symplectic quotient coming from a Hamiltonian
action of a compact Lie group is generally not a manifold (it is a
symplectic stratified space). Sjamaar defines a differential form on
this space as an ordinary differential form on the open dense stratum
that lifts and extends to a form on the original symplectic manifold.
He proves a de Rham Theorem, Poincaré Lemma, and Stokes’ Theorem using
this de Rham complex of forms. But these forms are defined
extrinsically. The symplectic quotient does, however, obtain a
natural diffeological structure, and one can ask whether the Sjamaar
forms are isomorphic as a complex to something intrinsic to the
diffeology.
I will show, under certain conditions, that this is the case: the
complex defined by Sjamaar is isomorphic to the complex of
diffeological differential forms. A useful consequence of this is
that one may use the Sjamaar differential complex in conjunction with
reduction in stages.
no comment as of now