ChapterZero

I visited CalTech this week, Monday through Wednesday (I’m on what passed for spring break), and spoke to 5 of 7 professors in the Applied and Computational Math Division, and maybe 7 graduate students. I also met two of the other prospective students (one of whom is from Rice, also in Houston). The atmosphere was a lot different this time around: when I visited through the GradPreview program, I was not very impressed by or attracted to the school, but this visit solidified my decision to attend. The problem with the impression I got from GradPreview is that it was unduely biased by the experiences of graduate students in the experimental sciences— of course they’re going to be bitter and resentful. But I will be doing math, which means not being stuck in the lab for the entire 4 or 5 years it takes me to get a PhD; the math grad students seem fairly happy with their schedules.

It seems that if I want to work with an ACM professor, as opposed to finding an advisor in one of the other departments, I’m going to have to develop an interest in PDEs. Candes, who I currently want to work with most, on image processing and signal representation stuff, is very popular with the current crop of grad students, who will be choosing their advisors in the year that I come in. So he probably will not have any money or time to be my advisor. The other profs are doing cool things: two of them are investigating multiscale methods, which deal with efficiently interfacing models of phenomena at different levels of detail. A simple example of this would be adaptive meshing for the solution of PDEs– where it is sometimes important to model a small region with almost quantum mechanical detail, and also important because of time considerations not to use the same level of detail in the majority of the regions considered. Multiscale methods seem relevant to a large range of problems, including protein folding, and other things like that. Then there’s one prof working on mathematical biology– specifically, designing dna base pair sequences to meet topological and dynamical constraints, for use as nano-machines. The mathematically interesting portion of his work involves optimization issues.

Anyhow, I found out Candes has a site on the cool stuff his group has been working on lately, called compressive sampling. The idea is that by minimizing total variation (difference in l1 norm), you can effectively reconstruct functions that have a sparse representation in some basis from a relatively small number of random samples/projections; because this gives a nice method of (lossy) compression, it’s sometimes referred to as compressive sampling. Simple idea, cool mathematics. Apparently it has all sorts of applications; one of the graduate students is even attempting to use compressive sampling to solve PDEs.