## Departmental PhD Thesis Exam – Jordan Watts

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

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.