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.

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