During the first year at Purdue, I was mainly interested in differential geometry. Since this isn’t very popular at Purdue, I was even thinking about to transfer. My interest in algebra was increasing as I was taking Ulrich’s commutative algebra, and Lipman’s abstract algebra.

Before the first summer, I was looking for some professor to take a book reading course. At first, Dr. Lee told me that she would be gonne for most of the summer. Then I emailed Yeung, Lempert, Donnelly and Catlin, but none of them is available. They are about all faculties around from whom I may learn some differential geometry. So I had to choose something different, and went to Lipman. He agreed to give me a reading course immediately, and later emailed me a book called “Algebraic and analytic geometry” By Aaron Neeman, which was published half year later. The results in the book were very hard. I couldn’t understand the main theorem, but still attracted by the topic. Hence I decided to work on Algebraic geometry.

Yeung is the first person I was thinking to work with. But unfortunately, he already had two students, and didn’t want any more. Under the advice of both Lipman and Yeung, I turned to Arapura, who was on sabatical for the whole year. Still I emailed him and he accepted! During my advisor’s absent, I spent most of the year reading Hartshorne, finished reading the first time around summer 08.

After my advisor came back at fall 09, he wanted me to go over the main part Hartshorne again with him. We spent about 3 months. In November, I took the advanced topic exam with Arapura and Lipman. Two weeks before the exam, Arapura gave a talk on the fundamental groups of compact Kahler manifolds. After that, he asked me some questions, one of them was “can every morphism between finite groups be realized as the induced morphism between fundamental groups from a holomorphic map between compact Kahler manifolds?” I was very interested and was able to solve this problem two days later. So right after the advanced topic exam, I started to work on this subject, with reading Voisin’s paper.

For more than two months, I was studying Kahler manifolds, and it’s topological structures. After the winter break, I switched the topic to vanishing theorems, because there is not a good problem in the previous subject proper for a graduate student. I spend most of the spring semester reading Esnault and Viehweg’s Lectures on vanishing theorems. Right before the Michigan workshop May 09, my advisor told me the specific problem he wanted me to think about. The approach he suggested was such a surprise to me, which also seems to be a long way to go.

Since this summer, I am thinking about either to learn some phisics or some number theory. We’ll see.

I should be able to learn (more or less) most of the stuffs(eventually):

1. Vanishing theorem (main job)
2. Positivity in algebraic geometry, Book by Lasarsfeld (working on it)
3. Hodge theory, Deligne’s work (working on it)
4. Minimal model program(the several fundamental theorems of MMP)
5. More transdental method in algebraic geometry
6. Intersection theory
7. Theory of algebraic curves
8. Stacks and moduli spaces
9. Proof of Serre’s GAGA
10. Algebraic K-theory

These are also interesting:

1. Resolution of singularity
2. Grothendieck’s standard conjectures
3. Weil’s conjecture
4. Index theory
5. Chern’s work on sphere bundle
6. Theory of differential operators

1. Kahler groups: If a group is isomorphic to a fundamental group of a compact Kahler(complex projective) manifold, the we call it “Kahler”(“projective”). In general, we can ask what are the necessary and sufficient conditions to be Kahler or projective. We know projective implies Kahler, since complex projective manifold is always compact Kahler. And we ask, “whether every Kahler group is projective?”
2. Kodaira’s problem: Kodaira proved every compact Kahler surface is equivalent to a projective surface via deformation. And Voisin constructed examples of dim>=4 compact Kahler manifolds, whose cohomology ring will never isomorphic to the one of any projective manifold, hence it is not equivalent to any projective manifold via deformation. But in dimesion 3, the problem is still open.
3. Combining the above two question together, we can also ask the necessary or sufficient conditions of a graded-commutative ring to be Kahler or projective, in the sense of 1.
4. Kahler morphisms: In the sense of 1, what morphisms between groups can be Kahler or projective?

The first three are more difficult and the last one seems more practical. Instead of settle down on one specific problem, I am wandering around and lead by my interest.