## Departmental PhD Thesis Exam – Michael Bailey

Everyone welcome.  Refreshments will be served in the
Bahen Math Lounge before the exam.

Friday, October 28, 2011, 3:10 p.m., in GB 244,
35 St. George Street

PhD Candidate:  Michael Bailey

PhD Advisors:  Marco Gualtieri and Yael Karshon

Thesis Title: On the local and global classification of
generalized complex structures
(http://www.math.toronto.edu/bailey/thesis.pdf)

Thesis Abstract:

We study a number of local and global classification problems in
generalized complex geometry, a relatively new type of geometry,
of which symplectic and complex geometry are special cases,
which has applications to string theory.

In the first topic, we characterize the local structure of
generalized complex manifolds by proving that a generalized
complex structure near a complex point arises from a holomorphic
Poisson structure. In the proof we use a smoothed Newton's
method along the lines of Nash, Moser and Conn.

In the second topic, we consider whether a given regular Poisson
structure and transverse complex structure come from a
generalized complex structure. We give cohomological
criteria, and we find some counterexamples and some unexpected
examples, including a compact, regular generalized complex manifold
for which nearby symplectic leaves are not symplectomorphic.

In the third topic, we consider generalized complex structures
with nondegenerate type change; we describe a generalized
Calabi-Yau structure induced on the type change locus, and prove
a local normal form theorem near this locus. Finally, in the fourth
topic, we give a classification of generalized complex principal
bundles satisfying a certain transversality condition; in this case,
there is a generalized flat connection, and the classification
involves a monodromy map to the Courant automorphism group.