Things I'm Learning

RIPS Midterm Presentations were a week ago, and after our midterm presentation and report, our team and sponsor agreed that the specific problem we had been working on was probably not going to lead anywhere. We'd been searching for closed forms for several values/properties related to a family of polynomials, and what we had found so far indicated that getting clean analytic results was unlikely. So we've pivoted to some more numerical and physics-related directions for the last couple of weeks of the project.


I was thinking through all of this, and I concluded that most of what I've learned falls into three categories.

"Not much is known" is quite different from "Not much has been done."

We thought we were coming into a problem on which relatively few people had worked. One of our tasks for the first part of the project was to conduct a more thorough literature review and find out how much people had looked at this family of polynomials and what they had found. It turns out that a small group of people had done significant work, simply without much success. Either the problem is very difficult, or it's simply not possible. (My team leans heavily toward the latter, though saying it's impossible is questionable because of some vagueness in the problem statement.)

"The steady state of mathematical research is to be completely stuck."

(Quote from this NYT article about Terry Tao)

I knew this quite well coming into this summer. But if there is anything this project has reinforced, it is that much of the story of doing mathematics is a lot of work resulting in not much progress. (This goes with the above, really.) Earlier this week, I walked into Mariette's office to check on something with her, and she looked at me and said, "I found the brick wall again." That's math research for you: banging your head against a brick wall, hoping at some point it will fall...so that you can move onto the next, even more exciting brick wall.

Math.

Without this project, I don't think I would have considered learning about orthogonal polynomials. Other than using a couple of nice families of them in PDEs, I'd never really seen them, and even then I didn't have to know anything about those families, just that they existed and were solutions to particular equations. But now I know so much about the theory of orthogonal polynomials, and I probably know more about Maxwell polynomials than all but maybe a couple dozen people in the world. In addition, for our new directions, I've been working on a numerical analysis part of the project. (I'm the one person on the team who knew almost zero numerical analysis... so of course I volunteered to do that bit.) In just a few days I've learned a lot about numerical analysis in general and quadrature and root-finding in particular.