Oct

18

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.

## no comment as of now