Home page of Yi Lin

Education

Ph.D.(August, 2004) Cornell University, Mathematics
Advisors: Professor Reyer Sjamaar

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 have been an instructor with the University of Illinois and the University of Toronto. I have taught a variety of undergraduate courses ranging from introductory calculus to upper level probability and geometry courses.

  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 extend the whole Kirwan package to Hamiltonian actions on generalized complex manifolds.

Publications.

1: The equivariant cohomology theory of twisted generalized complex manifolds, math.DG/0704.2804, Comm. in Math. Physics, 281 (2008) 469 - 497.

2: The log-concavity conjecture for the Duistermaat-Heckman measure revisited, International Mathematics Research Notices, (2008) Vol. 2008, article ID rnn027, 19 pages, doi:10.1093/imrn/rnn027. 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

Symplectic Seminar
FIELDS INSTITUTE
Arxiv

---

Professional Affiliations

American Mathematical Society

---