The goal of our project is to prove Conjecture 6.5 located in the paper titled “Curve Neighborhoods of Schubert Varieties” by Anders Buch and Leonardo Mihalcea.
- Week 1:
- Met with my advisor Professor Anders Buch twice this week (once with his PhD student Sjuvon Chung). Reviewed the material assigned at the end of the school year on Grassmannians and Schubert varieties. Discussed alternative definitions of the Schubert variety and how to show equivalence. Professor Buch showed how to define a line in the Grassmanian, and began talking about line neighborhoods and curve neighborhoods. Began reading about root systems and Weyl groups.
- Week 2:
- Gave first presentation about the project with my partner Chi-Nuo Lee. Met Professor Buch and Sjuvon Chung and discussed a curve of degree 2 or more in the Grassmannian, the Plucker embedding, as well as the tangent plane. Began to relate root systems to Grassmannians.
- Week 3:
- Met three times with Sjuvon this week since Professor Buch was away. Discussed how to describe curves of some degree d in the Grassmannian by first looking at curves in projective geometry. Wrote up the proofs to some exercises involving Schubert varieties and curve neighborhoods with Chi-Nuo.
- Week 4:
- Met with Professor Buch. We discussed the Weyl group of type A and flag varieties and what a Schubert variety is in some flag variety as opposed to a Schubert variety in the Grassmannian. Talked about how the dimension of a Schubert variety can correspond to either a Young diagram in the case of the Grassmannian or the length of a Weyl group element in a flag variety. We were assigned some reading on the Hecke product of Weyl groups in a paper titled “Curve Neighborhoods Of Schubert Varieties” by Anders Buch and Leonardo Mihalcea.
- Week 5:
- Met with Professor Buch on Monday. We're moving closer to understanding the conjecture. This week our main focus was to see how the construction of each Schubert Cell can be generalized in flag varieties and partial flag varieties. By looking the stabilizer of a certain element in the partial flag variety, we can now understand the Grassmannian as a special case of the partial flag variety. We learned how the indexing works for Schubert Cells in partial flag varieties, and how such indexing relates to our special case of Grassmannian. We achieved that by looking at permutation matrices, and choosing representatives for cosets of permutation matrices that does not change the flags. We hope to finally understand the conjecture in the next week.
- Week 6:
- Met with Professor Buch and talked a bit about homology and the pushforward. Discussed the conjecture in more detail and was assigned sections of the paper to read in order to tackle the problem. After reading the paper, we decided to approach the conjecture based on the type of Weyl group. We will focus our efforts on the simplest simply laced root system which is of type A.
- Week 7:
- Met with Professor Buch and showed him our proof for the conjecture with Weyl groups of type A. His initial thoughts were that the proof was correct. We have a proof also for type D and will now consider variations of the conjecture, specifically what occurs in the non-simply laced case.
- Week 8:
- We have verified our proof of the conjecture with Professor Buch, we now seek to classify all P-cosmall roots for type A,B,C, and D root systems. We also have begun writing up our paper, which we hope to have published. We will continue to explore a more general version of the conjecture, but so far we have not found anything.
- Cox, D. A., Little, J. B., & O'Shea, D. (2007). Ideals, varieties, and algorithms: An introduction to computational algebraic geometry and commutative algebra. New York: Springer.
- Humphreys, J. E. (1972). Introduction to Lie algebras and representation theory. New York: Springer-Verlag.
- Curve Neighborhoods of Schubert Varieties
- Notes on Grassmannians
I will study spherical harmonics. Consider the unit sphere in 3 dimensions. We can consider the tangent space of the sphere. Call this space V. Then if we look at V tensored with itself k times, we can examine the invariant space that is stabilized by the special orthogonal group SO(2). I will attempt to characterize the set of functions in the orbit on differential operators in this space acting on eigenfunctions of the Beltrami-Laplace operator.
- Week 1:
- I met with my advisor Anders Buch. We recently found a generalization for a project that we worked on last year so we discussed the proof of this new theorem and I revised my parper that I worked on with Chi-Nuo. I met with Fei Qi and we discussed the new problem to work on for this year's REU. He recommended me some background reading in Tu's Introduction to MAnifolds and I began exploring the problem by proving a correspondence betwen eigenfunctions of the Laplace on the sphere and Legendre polynomials. I ended the week by setting up my website and creating a presentation.
- Week 2:
- Met with Fei Qi to discuss the solutions of eigenfunctions on the unit sphere. I had a result relating to the general Legendre equation which I was unaware of. I will research the general Legendre equation and attempt to characterize the eigenfunctions.
- Week 3:
- Talked to Professor Buch again about the paper from lsat year. He mentioned the possibility of a geometric argument involving root strings, which might simplify one of our proofs. Had a weekly meeting with Fei Qi. We cleared up some ambiguity surrounding the boundary conditions in solving for spherical harmonics. There is an argument based on the Dirac operator, which he will explain to me at our next meeting. My next task is to determine parallel sections in the pspace we originally discussed. Then I will act on some particular eigenfunction and examine the orbit of the action.
- Week 4:
- On Monday, met with Fei Qi and went over some terms in differential geometry that I did not fully understand. We discussed how to compute the parallel section that we are looking for. I will continue reading my reference text in order to gain a better understanding of this feild.
- Week 5:
- Professor Buch contacted a German Ph.D. student who also independently proved Buch's conjecture as part of his thesis. Met with Fei Qi on Wednesday and discussed more about hte covariant derivative and how exactly we hsould define the action. Provided me with another reference book on Riemannian geometry that I will begin reading.
- Week 6:
- Looking over the thesis that Professor Buch forwarded to me. It seems that the work I did in the REU last year is different enough that we can publish, but there are still some sections that I need to check carefully. Also still trying to figure out how to compute the action that Fei Qi defined for me the previous week, but not making much progress. Continued to read the books that Fei Qi gave me in order to gain a better understanding of geometry and making more progress after being stuck on certain parts of the book for quite some time.
- Week 7:
- Began some computations on the action of the tensor fields. There does not seem to be any immediate pattern so I might have to write a Maple program to carry out the computation. Gave my final presentation on the project. Met with Professor Buch so that he could explain some of the concepts defined in the paper.
- Week 8:
- It turns out that the action previously defined is not exactly correct as when you act by a certain tensor, you do not get multiplication by the eigenvalues, which is what you would expect to get. Met with Fei Qi and he told me to read Tu's books up to the section on Principal Bundles, which should give me more intuition to correctly define the action.