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 Arthur Wang. 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 Arthur.
- 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