Community

The past couple of  weeks have made me think about communities of which I've been a part. There are a lot of them, and often it's the people that end up being most important to me about a place or experience. Here are a few recent moments that have really highlighted community for me.

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.

Approaching Hard Problems: Math and Engineering

My friend Amelie and I have been talking about our summers, and in particular the differences in our research came up. She's on an interdisciplinary engineering team working on blimps, and I'm doing math research that has many applications but isn't really applied in and of itself. 

It turns out that my team is working on a problem that is harder than we expected. We're looking for a number of different pieces, and we've found that having any one of them would allow us to find all of them without too much work. The problem is finding one, and we're a bit stuck. There are directions that we're exploring, and we've confirmed a lot of the work that we've found in literature, but at this point it doesn't feel like we're going to be able to make much progress, and we'll probably end up pivoting to a related but more approachable problem.

Amelie's reaction when I told her this was that math research seemed pretty scary, that in her kind of research, maybe there's one way she would prefer to do things, but really she could come up with six different ways. If something doesn't work, there are other ways to accomplish the same task. But when we're stuck, we don't necessarily have that option. We try to find ideas for new approaches or information from doing more literature review and from discussions with other people, but sometimes that doesn't lead to anything helpful.

We're looking for closed forms of certain expressions, and we're not even sure they exist. We know that if they do exist, they're far from elegant, but we don't really have ideas for proving the closed forms don't exist, either. So we're reading, trying to follow lots of trails, and talking to our liaison a lot for suggestions on directions to go right now and about possibilities if this is just too intractable.

A Few Summer Pictures

I've been away from Olin for almost a month and a half, so I thought I'd post about what I've been up to! Pictures below the fold.

Thoughts on Spring 2015 Activities

Here's a little bit about what I did this semester that wasn't related to classes or math!

Thoughts on Mechanics of Solids and Structures and Math Activities

My fourth class this past semester was Mechanics of Solids and Structures, or MechSolids, which is a mechE requirement. I also did a lot of math related activities. Compared to last semester, I didn't feel like I did very much math this semester, but what I did do was pretty varied.

It Flaps But Does Not Fly: Thoughts on Mechanical and Aerospace Systems

MechAero is a course I should not have taken.

Part of the mechanical engineering degree is a mechE elective, and MechAero counts as such an elective. It isn't taught very often, and I was excited because I thought, based on the course description, that I would get to do a lot of analysis, especially analysis related to fluid mechanics. Instead, the course was almost entirely a mechanical design class with little to no support for learning to do mech design.