| Student: | Heman Gandhi |
|---|---|
| Office: | CoRE 444 |
| School: | Rutgers University |
| E-mail: | hemangandhi@gmail.com |
| Project: | Schubert Calculus |
| Mentor: | Anders S Buch |
Project Description
Schubert calculus is the study of lines and curves: we look into how many lines can intersect a set of lines. The cohomology of Grassmannians is immediately tied in through the projective plane, so that Schubert varieties can be shown to give solutions to the problems with lines and curves. This, in turn, brings in the Schur functions. Studying vector bundles, instead of just the varieties, gives rise to Grothendieck polynomials. Grothendieck polynomials are combinatorially explained by set valued semi-standard tableaux. So, the lines and curves can be explained using boxes, partitions, and sets filling them. We hope to find an algebraic basis for the Grothendieck ring.
Weekly Log
Week 1The week started with moving in and meeting all the fellow researchers. It was fun to hear of everybody's unique backgrounds and their exciting projects! The next day we dug in: after orientation, we met professor Buch and heard a lot of the background of our project: how projections made "counting the lines intersecting a set of lines" relate to subspaces (planes) in higher dimensions, and how that in turn relates to tableaux through Schubert calculus. We continue to work on the exercises and fully understanding and learning all that we heard. We also have an introductory presentation due... :/
Week 2After wandering the wild woods of cohomology for a few more days, we were introduced to K-theory. Luckily, we were not expected to delve into it too much - only to understand that it's like a cohomology, but more complex (so really, cohomology is a special sort of K-theory, perhaps). The abstractions aside, we were officially told our conjecture and then the kind of tools required to solve it. Essentially, from cohomology, there is a Littlewood-Richardson rule that lets one multiply Schur functions. There is a K-theory analogue using set-valued tableaux. We use this rule to multiply polynomials and our conjecture relates to a basis for these polynomials. Here, we get lucky again: it was recommended that we write a program to multiply and have been coding! We also have a few sub-conjectures and sub-sub-conjectures (we didn't go any deeper, don't worry).
Week 3The code got faster: I switched to Haskell and got a 3x speed-up. The algorithms are more optimal too, with stronger bounds and more stuff we figured out - there are many handy tricks. Now I'm looking to convert the solver into a linear-algebra-based algorithm. Working with matrices is tricky and being sure of the correctness of the algorithm is hard - there are a few uncertainties as we have very few facts and equations to work with.
Week 4
After an unproductive Monday, on Tuesday, we solved the conjecture for all "L" shapes. This impressed the professor
and emboldened us. By Thursday night we were on the cusp of a formula for 2-row shapes and on Friday we proved it.
I find it surprising that we're able to produce formulas. I thought some sort of contradiction would drive us to
a non-constructive proof. Additionally, we're using fewer rectangles than I thought we would be.
Also, this Wednesday, we had the fortune of going to IBM's research center in Yorktown. It was a lot of fun (though
some of the talks weren't). I got to see a quantum computer and understand it a little. This inspired me to check out
their online playground and Python library.
We were stuck on 3-row shapes for the whole week. (We still are.) There are some difficult cases since when row lengths match up, we are powerless to use our older techniques. We tried and failed on various inductive ideas. We think we can reduce most shapes to ones there the first and the second row match up. Then we reach an ugly case that is a bit scary.
Week 6We are still stuck on 3 row shapes, technically. Instead of focussing on this, we have began looking at generalisations. If all the row lengths differ, we have found a way to reduce the shape down to a case where some rows match up. I implemented some sort of inductive solver in Haskell so that we can look at the base cases of our inductions quickly.
Week 7We presented all the above findings and learned about the work of our colleagues. We have began experimenting with other shapes since three row shapes are rapidly seeming intractable. We are also looking at multiple different ways to look at L shapes in the products.
Week 8
We wrote most of the final paper. We learned more about the impacts of the conjecture and more about the underlying complexities of K-theory. We also learned that the conjecture has a cohomology analogue. We also finally know how to handle more 3-row shapes and have them done for short third rows. Hopefully, generalization will reach us soon.
Week 9
We finished. The paper, the program, but not really 3-row shapes. There is a magic formula and no proof. There were some hearfelt goodbyes as the program ended. The paper writing was a lot of fun - we proved (among our results) that we can actually write a math paper and that felt very good. Explaining this has led me to understand the research better and really contextualize our findings and even the conjecture. This was an awesome summer and I can't express enough gratitude to the DIMACS REU program.
Presentations
- Coming soon...