I will probably be in Boston from May 27 through June 1. My original reason for going there has evaporated, but because of how it fits in with other travel, it makes little sense for me not to spend at least a couple days in Boston regardless. Any suggestions for what I should do while there?
The workshop this past week was enlightening. Most of the attendees
had backgrounds in dynamical systems; I know something about
dynamical systems, but it's by no means something I know well. But
I do have some common background with lots of other applied
mathematicians, namely the family of subjects that tend to fall
under the heading of
analysis: ODEs, PDEs, functional
analysis, complex analysis, geometry, some mathematical physics. So
I was able to generally follow most of the discussion.
A big theme of the week was computing or approximating or bounding the spectra of different types of linear operators. There is, I think, a very interesting tradeoff in this sort of analysis. It comes up in other sorts of analyses I've thought about, too. On the one hand, you can pose your problem as something that takes place on an unbounded domain, and get something that's linear (though linear eigenvalue problems actually involve nonlinear equations). The linearity of the problem is nice, but it's sometimes harder to deal with problems on infinite domains (certainly it's harder numerically). The trade is that you can take the same problem and reduce it to something that lives on a nice, bounded, compact set, which is good for computation... but then your problem becomes nonlinear.
Well, I like nonlinear eigenvalue problems, but I don't like them so much that it keeps me from doing tricks in order to turn them into linear problems as quickly as I can. Computers only handle finite things, so an unbounded mesh is sort of a problem, and so I can't deal with the real linear problems. But I can trade off how large a discrete problem I'm willing to look at with how nonlinear a problem I'm willing to tolerate. In fact, I can make that trade in a controlled way: starting from a small, fully nonlinear eigenvalue problem, I can introduce a sequence of approximations, each of which is a linear eigenvalue problem. My approximations get better and better, but at the expense of introducing more and more dummy variables. In the limiting case, at least for some problems, I can show that the approximations tend to the actual nonlinear eigenvalue problem... but they also get infinitely big.
This is pretty cool, but the really cool thing is that in similar contexts, I've spent a lot of time thinking about how to get the most out of my added variables. So not only can I make good approximations while remaining linear, I can make good approximations that aren't too big. At least, I can do that for some problems.
This coming week I will be in Pennsylvania for another meeting (this one on numerical linear algebra), and I'll return to California around the middle of the following week. I'll be glad to return to a more ordinary schedule. Traveling is okay, though I don't enjoy it nearly so passionately as some of my friends do; and academic meetings can be extremely interesting. But they're also very fast and intense, and I get worn out after a while. It's also an unfortunate fact that talking about ideas is not the same as developing those ideas, so I very rapidly develop a backlog of things I'd like to play with but for which I've not yet had time.
- Currently drinking: Coffee