Shreya Sinha | DIMACS REU 2024

Week 1 (5.28 - 6.2)

I met with Prof. Tobasco and Sam, and went over the broad outlines of the project as well as got started with easier cases of our problem. Prof. Tobasco recommended the book Convex Analysis and Nonlinear Optimization by Jonathan Borwein and Adrian Lewis. I read + took notes through Chapter 3.1. I also started creating my REU site, as you can see! Finally, I solved a nice case of one of the problems (the annulus), and started planning to tackle the other cases.

Week 2 (6.3 - 6.9)

This week was mostly just readings from Borwein & Lewis, as well as from the 2022 ARMA paper by Prof. Tobasco in order to learn more theory and justification behind why my caculations were valid.. I tried two more cases: the shifted annulus, where the hole is no longer centered at the center of the disc, and the disc with sections of differently signed curvature. I made some progress with both, but not as far as the original annulus. Prof. Tobasco and I went over the annulus calculations, and with further justification for some details, it seems like our calculations hold.

Week 3 (6.10 - 6.16)

This week Prof. Tobasco and I went over the physics behind the shell problem, and how he turned it into the dual optimization problem using convex analysis techniques. I reviewed the first three chapters of Introduction to Real Analysis by Robert C. Gunning to brush up on the basics of real analysis. We also found out from Prof. Joseph Paulsen at Syracuse University who runs experimental versions of the thin shell problem that our predictions for the annulus seemed correct! We also had an idea for the shifted annulus and the disc with sections of differently signed curvature, so that is my new focus.

Week 4 (6.17 - 6.23)

I read the Convex Analysis chapter (12) from the book Linear Algebra and it's Applications by Peter D. Lax. I also heavily focused on the disc with sections of differently signed curvature, as I had a few promising ideas. Some of the toy examples I worked with turned out to be correct, but when I tried to generalize my theory, it fell apart (the humbling joys of research!). Prof. Tobasco suggested a different direction to approach it, which also seems promising and is my new focus. I also met Prof. Paulsen over Zoom, and I learned more about the experimental process of creating and testing the wrinkling patterns of thin shells, as well as possible limitations to the process and thus to what examples and theory we can test. Finally, I plan to attempt the shifted annulus problem in more depth towards the beginning of next week. In less technical news, the we (the REU participants) had a very fun culture day ice cream social and potluck (6.20) :)

Week 5 (6.24 - 6.30)

I coded up the disc with sections of differently signed curvature in Python to find the optimizations of specific test cases and try to establish a pattern. I found some interesting results, but nothing that I was able to generalize immediately. In our meeting, Prof. Tobasco introduced a conjecture that would generalize a major result of his 2021 paper to the more general scenario we were considering. After some interesting formalizations, we found one that seemed promising. My goal for the rest of the week is to prove the latter half of the conjecture.

Week 6 (7.1 - 7.7)

It's been proved! After fruitless hours of bounding, it just so happened that an equality statement we needed was exactly true in the case we were working in. I learned an important lesson: always double check the statements of theorems in textbooks, not online; many hours will be saved this way :) The next goal is to try to apply this to the regular annulus case and see if what our calculations from earlier last month agree with finding a solution via our newest result.

Week 7 (7.8 - 7.14)

Week 8 (7.15 - 7.21)

Week 9 (7.22 - 7.28)