Last Updated: 12/14/2011
I received my Ph.D in August, 2004 from Cornell University, with Reyer Sjamaar as my
thesis advisor. In the academic year 2004-2005, I was a visiting assistant professor
at the University of Illinois at Urbana-Champaign; in the academic year 2005-2008, I was a Postdoctoral Fellow in the Department of Mathematics at the
University of Toronto, where my mentors are Lisa Jeffrey, Yael Karshon, and Eckhard Meinrenken .
Currently I am a tenure track assistant professor in the Department of Mathematical Sciences at Georgia Southern University.
Curriculum Vitae
Brief Description of
Publications
Education
Teaching
I began teaching in 1999 as a teaching assistant at
Cornell University. Since then, I have had ten semester-long appointments as a
teaching assistant at Cornell University; and I had been an instructor
with the University of Illinois and the
University of Toronto before I came to Georgia Souterhn University. I have
taught a variety of undergraduate courses ranging from introductory calculus
to upper level probability and geometry courses.
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MATH 2242
MATH 3230
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. However, my recent work on symplectic
Hodge theory and primitive cohomology classes has also aroused my interests in
geometric measure theory.
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. In collaboration with Tom Baird, I extend the whole Kirwan
package to Hamiltonian torus actions on generalized complex
manifolds.Very recently, I proved that there is a
Poincar\'e duality between the primitive cohomology and homology on any
compact symplectic manifold with
the Hard Lefschetz property. For projective K\"ahler manifolds, this provides a new
geometric interpretation of primitive cohomology classes which is very different
from what algebraic geometers had before. As an application, I gave a
rather satisfactory answer to an open question asked by Victor Guillemin on
the symplectic Harmonic representatives of Thom classes. Among other things,
I extended the Whitney's notion of flat chains to symplectic manifolds,
and used it to give a geometric construction of the symplectic Hodge
star operator. This reveals an unexptected intrinsic connection
between the symplectic Hodge theory and the geometric measure theory. I
intend to explore this connection further in a series of follow-up
works.
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Publications.
1: Topology of generalized complex quotients
I, with Tom Baird, 33 pages, the Journal of London Mathematical Society, (2010) 81 (3):
573-588.. PDF
2: Generalized complex
Hamiltonian torus
actions: examples and constraints, with Tom Baird, math.DG/0904.1178,
19 pages, the Journal of Geometry and Physics, Volume 60,
Issue 10, October 2010, Pages 1539-1557. PDF
3:Non-Kahler symplectic manifolds with toric
symmetries,
with Alvaro Pelayo, 12 pages, the Quarterly Journal of Mathematics, the
Quarterly Journal of Mathematics, (2009)
doi: 10.1093/qmath/hap024. PDF
4: The equivariant
cohomology theory of twisted generalized complex manifolds, Comm. in Math.
Physics, 281 (2008) 469 - 497.PDF
5: The log-concavity conjecture for the Duistermaat-Heckman measure revisited,
Preprint, Internaional Mathematics Research Notices,
(2008) Vol. 2008, article ID rnn027, 19 pages, doi:10.1093/imrn/rnn027PDF
6: Generalized geometry, equivariant
$\overline{\partial}\partial$-lemma, and torus actions, math.DG/0607401, the
Journal of Geometry 6nd Physics, 57 (2007) 1842-1860. PDF
7: Symmetries in generalized K\"ahler geometry, with Susan Tolman,
Commun. Math. Phys., 208 (206) 199-222. PDF
8: Examples of Non-Kahler Hamiltonian circle manifolds with the
strong Lefschetz property, Advances in Math., 208 (2007), issue 2, 699-709.
PDF
9: Equivariant symplectic Hodge theory and the $d_G$,
$\delta$-lemma, with R., Sjamaar, J. Symplectic Geom. 2 (2003), no. 2,
267-278. PDF
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