Teaching
Statement
Education
Teaching
MATA35 Calculus II
for the Biological Sciences
MATC25 Classical
Plane Geometries and their Transformations
MATA27 Introduction
to Optimization
Research Interests
I am interested in symplectic Geometry, generalized complex
geometry, and their connection to Mathematical Physics and Lie Theory. My
research so far mainly concerns the study of symmetry in symplectic and
generalized complex geometries.
In my thesis work, I studied symplectic Hodge theory and the Hard
Lefschetz property. Using the symplectic Hodge theory, I constructed
a
very simple proof of an improved version of the Kirwan-Ginzburg equivariant
formality theorem. In addition, I constructed the first counter examples to an
open question raised by Kaoru Ono and Reyer Sjamaar of whether the Hard
Lefschetz property survives the symplectic reduction.
After I received my Ph.D, I started my work on generalized complex
geometry, an area initiated by Nigel Hitchin a few years ago. Jointly with
Susan Tolman I extended the notion of Hamiltonian action and Marsden-Weinstein
reduction to the realm of generalized complex geometry. As a first
application, we worked out explicit constructions of bi-Hermitian structures
on many toric varieties whose existence was only conjectural before. Recently,
it has been shown by Kapustin and Tomasiello that the conditions that Tolman
and I used to define generalized Kahler quotients are exactly the conditions
in physics for general (2,2) gauged sigma models. In a series of follow up
papers, I studied the equivariant cohomoloy theory for Hamiltonian actions on
twisted generalized complex manifolds.
As an application, I considered the Hamiltonian torus actions on twisted
generalized Calabi-Yau manifolds and extend to this case the
Duistermaat-Heckman theorem for the push-forward measure. More recently, in
collaboration with Tom Baird, I established the Kirwan surjectivity results
for generalized complex quotient. I refer you to my publication list and
research statement for more information of my research.
Publications.
1: The equivariant cohomology theory of twisted generalized complex
manifolds, math.DG/0704.2804, accepted by the Comm. in Math. Phy. for
publication. PDF
2: The log-concavity conjecture for the Duistermaat-Heckman
measure revisited, Preprint, math.SG/0703297, accepted by International
Mathematics Research Notices. PDF
3: Generalized geometry, equivariant
$\overline{\partial}\partial$-lemma, and torus actions, math.DG/0607401, the
Journal of Geometry and Physics, 57 (2007) 1842-1860. PDF
4: Symmetries in generalized K\"ahler geometry, with Susan Tolman,
Commun. Math. Phys., 208 (206) 199-222. PDF
5: Examples of Non-Kahler Hamiltonian circle manifolds with the
strong Lefschetz property, Advances in Math., 208 (2007), issue 2, 699-709.
PDF
6: Equivariant symplectic Hodge theory and the $d_G$,
$\delta$-lemma, with R., Sjamaar, J. Symplectic Geom. 2 (2003), no. 2,
267-278. PDF
Preprints.
1: Examples of Hamiltonian actions on twisted
generalized complex manifolds, Preprint, 13 pages, August 2007.
2: Geography of non-K\"ahler symplectic torus actions,
with Alvaro Pelayo, 13 pages, submitted, August 2007. PDF
3: $d_G$, $\delta$-lemma for equivariant forms with generalized
coefficient, 19 pages, submitted.
4: Cohomology of generalized complex quotients I, with Tom
Baird, arXiv: math.DG/0802.1341.
Recent talks available in pdf:
1: [PDF]
Hamiltonian actions on generalized complex manifolds and equivariant
$\overline{\partial}_G\partial$-lemma, CMS 2006 winter meeting, Toronto,
Canada.
2: [PDF]Hamiltonian
symmetry in generalized complex geometry, November, 2006, University of
Western Ontario
Mathematical Links
Professional Affiliations