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 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.
                    
                      MATH 2242

                      MATH 3230

            

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        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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        Preprints

1: Currents, primitive cohomology classes and symplectic Hodge theory, Prepreint, 52 pages, arxiv: 1112.2442. PDF

2: $d_G$, $\delta$-lemma for equivariant forms with generalized coefficient, 19 pages, submitted.

 

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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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