General Information

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Project Description

In algebraic geometry, we are interested in flag manifolds, such as the Grassmanians. The goal is to work with certain flag vatieties and to perform computations in their cohomology rings using tableaux.

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Weekly Log

Week 1:

I met my mentor this Wednesday. He gave me a high-level overview of the motivation for my project. It has relevance for the cohomology rings of certain varieties. Then he introducted some of the practical aspects of what I'm working with, namely the symmetric polynomial ring R = Z[x1, ..., xn]. He gave me some notes he wrote on the Schur Polynomials, which I started reading yesterday.

Additionally, I'm working on verifying that the Elementary Symmetric Polynomials form a basis for the ring of all symmetric polynomials. So far, I've proved the cases n=2 and n=3 using computational induction. But I don't think this approach will work for arbitrary n. I think I'll need a more theoretical approach for that.

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Week 2:

I successfully verified that the Schur polynomials form a basis for the set of symmetric functions. I figured out how to do this while waiting for my mentor to arrive after I arrived early to a planned meeting. So that's good. During the meeting itself, Prof. Buch gave me futher overview of the area of research. He also introduced me to root systems.

Later in the week, I tried reading a paper on Grassmanians, with very little success. But I successfully started reading Ch. 3 of Humphreys' "Lie Algebra algebra and representation Theory". THis chapter covers root systems. I think this reading, like the notes on Schur polynomials, will go very well.

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Week 3:

I finished working through root systems. I look forward to using them in the future.

I've begun working through Young Tableaux in depth. I'm working on understanding the Pieri Formula in the abstract, how it follows from Tableau rules.

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Week 4:

This week, I worked through the Pieri formula and the Littlewood-Richardson formula. These describe the multiplication of Schur functions, as well as the computation of skew Schur functions.

I also started diving deeper into Flag Varieties. I'm applying Root Systems and their Weyl Groups to understand certain varieties.

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Week 5:

I've begun working on computations in varieties. For now, I'm working on computing cohomology ring multiplication tables in varieties corrosponding to root systems of rank 2. I've had some success with A2 and A1xA1. But I need to get a better understanding of root systems/ Weyl groups to use more tricks in these computations. I'll spend time next week looking at Humphreys' Reflection Groups and Coxeter Groups. My advisor says this should help.

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Week 6:

I've finished the relevant readings from Humphrey's RGaCG. I'll move on to computations next week. I do anticipate being busy with other REU activities, but I should still have plenty of time.

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Week 7:

The computations had some initial success, but I have not been able to get anything serious done. I did not do the computation for B2 or G2.

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Week 8:

I still have not made progress on the computations. Unfortunately, I have to accept that I've done what I can. The rest of my time will be focused on wrapping up and solidifying what I already learned, while writing my final report.

This week was still quite productive for me. We had a talk on math puzzles that wsa very interesting and that I've already discussed with my friends outside the REU. We also had final presentations. It was really interesting to see what everyone else has done. Plus the REU director Lazarus told me my presentation was good. I'm happy how that went.

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Week 9:

I spent this week as planned, reviewing what I've learned and writing my final report. While I did not do any concrete research, I got exposed to research-oriented learning. I learned a lot of new math in general, which is always a good thing. I'm glad I did this REU. So long!

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