A very partial list of names whose papers I often try to read goes something like (in random order): A. Floer, S. Donaldson, , M. Gromov, R.R. Fintushel-Stern, Kronheimer-Mrowka, D. McDuff, C. Taubes.
Some other names I struggle to read are: A. Floer, S. Donaldson, C. Taubes.
I always have a soft heart for the hyperbolic 3-manifold and the knot, I guess being at OSU helps. How I wish I could solve some 3-manifold problems!
My current three wishes in geometry are:
(1) Prove the Atiyah conjectures.
(2) Finding product strucutre for the instanton Floer homology.
(3) Prove the the axiomatic theory for Floer instanton homology.
| My favorate results | Theorem 1 Ther are instanton Floer homology of rational 3-spheres which are dependent on the
perturbation with fixed Walker's correction term Theorem 2 There exists an natural isomorphism between the instanton Floer homology and the symplectic Floer homology of homology 3-spheres |
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| Publications List | Here is everything I wrote that were published or intended to be published. I wish the list were longer. But it could have been shorter. You can download most of these papers. |
| Most Recent Preprints | Want to know whether I've been lazy lately? The length of the list is inversely proportional to the time I spent surfing the net. |
| Three manifolds and Knots | When I graduated from MSU, I felt very satisfied to write a thesis about homology three spheres, with the hope that as I grew older, my manifolds would get intomore complicated. Horrified was I to discover recently that my manifold has actually become "easier" (like a knot). Here are all I wrote about dimension three. It is often asked what more information one can say about the Floer homology. |
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| Instanon Floer homology | The 3-sphere is intriguing. The Instanton Floer homology initialized by Taubes' work is powerful, but too hard to compute or say something about 3-manifolds. I'm proud that my thesis advisor brought me into this field. Although there may be one day some kind of ``Seiberg-Witten'' on 3-manifolds sweeping many 3-topology problems, I am still thinking the rich structure on the instanton Floer homology. Jointly with Ronnie, I believe that one day I can come up an application of Floer homology to 3-manifolds. |
| Symplectic Floer homology | Why symplectic? Once people claimed that the world is symplectic. The 4-dimensional symplectic manifold is hot topic now, so is 3-dimensional contact manifold(a twin of symplectic in odd dimension). Many results about symplectic manifolds do shows some special features, especially Aronald's conjecture and Gromov's squeezing theorem. The original Casson's invariant is of symplectic approach, some knot information also can be obtained from the symplectic topology. I wrote these, and hope more to come. |
| Unclassified | These are almost random thoughts, at least on the surface. But they are related to some of the above subjects. |
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