Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2010. / Cataloged from PDF version of thesis. / Includes bibliographical references (p. 65-66). / In this thesis, we introduce the odd dimensional symplectic manifolds. In the first half we study the Hodge theory on the basic symplectic manifolds. We can define two cohomology theories on them, the standard basic de Rham cohomology gheory and a basic version of the Koszul-Brylinski-Mathieu 'harmonic' symplectic cohomology theory. Among our main results are a collection of examples for which these cohomology theories don't coincide, and, in fact, for which the usual basic cohomology theory is infinite dimensional and the symplectic cohomology theory is finite dimensional. On the other hand, we prove an odd version of the Mathieu theorem and the do-lemma: the two theories coincide if and only if a basic version of strong Lefschetz property holds. In the second half, we discuss the group actions on odd dimensional symplectic manifolds. In particular, we study the Hamiltonian group actions. Finally we use the Local-Global-Principle to prove a convexity theorem for the Hamiltonian torus actions on odd dimensional symplectic manifolds. / Ph.D.
| Identifer | oai:union.ndltd.org:MIT/oai:dspace.mit.edu:1721.1/60189 |
| Date | January 2010 |
| Creators | He, Zhenqi, Ph. D. Massachusetts Institute of Technology |
| Contributors | Victor W. Guillemin., Massachusetts Institute of Technology. Dept. of Mathematics., Massachusetts Institute of Technology. Dept. of Mathematics. |
| Publisher | Massachusetts Institute of Technology |
| Source Sets | M.I.T. Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 66 p., application/pdf |
| Rights | M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission., http://dspace.mit.edu/handle/1721.1/7582 |
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