Physicist Amok

Here’s a post which poses some questions about mathematics training for biologists. I’m close enough where I figured I’d try to answer:

1. Are you a biologist, if so what kind? Sort of. My PhD will be in biology, on the response of mycobacteria to antibiotics. My undergraduate degrees were in physics and math.
2. What math did you take in college? Calculus through multivariable, ordinary differential equations, a two semester course on math methods for physics students covering special functions, complex analysis through contour integration, and some partial differential equations, real analysis through the last course they shove the doctoral students through, the probability course that comes just after, a bit of functional analysis, and a two semester graduate sequence in algebra.
3. What math do you or have you used? My image analysis requires some partial differential equations, a lot of calculus, and a bit of differential geometry. I do some theoretical biology on the side which particularly takes the probability and random processes, and partial differential equations as the limiting cases of the former. My bench work itself doesn’t take more than highschool algebra, but designing and analyzing my experiments requires statistics, which I’ve been teaching myself. Before this, when I was training to be a mathematical physicist, I obviously used everything I knew and it wasn’t enough.
4. What math do you wish you’d studied? I wish I had actually endured the second year of graduate algebra, and a graduate course on combinatorics. A couple semesters of statistics would have saved me a lot of time now, but I can fill in the gaps. I really wish I had taken serious courses on differential geometry and topology.
5. How do you use math in your job (or research)? See above.

I think a case could be made that biologists should take all the math required to take a full scale mathematical statistics course including multivariate and nonparametric methods, and a heavy dose of experimental design. Let’s see: real analysis through measure theory and Lebesgue integration and Stokes’s theorem on manifolds, a hefty course on probability from the axiomatic basis to the beginnings of random processes, and then blast through statistics because a t-test would only require about ten minutes and a homework problem, and so would a U-test. This is four years (two of analysis, one of probability, one of statistics), but it’s only one course a term.

Admittedly, biology departments would suddenly have as many undergraduates as physics departments.

madhadron :: May.24.2007 :: biology, mathematics, teaching :: 1 Comment »

Being a physicist at the Rockefeller University — and anyone who isn’t a biologist or a biochemist is a physicist around here — means that any mathematical problem that arises lands in your lap. This is both enlivening and rather uncomfortable. Of late, many of the problems have been statistical, which is particularly uncomfortable: I’m extremely good with probability in all the gory, measure-theoretic details, and even somewhat conversant with random processes, but I never did any statistics.

So I’ll learn. First, I need decent books, but I don’t know what books are decent until I know something about the field. To pull myself up by my bootstraps, I go to the relevant section of the library, and start pulling books off the shelves, flipping through them, and trying to figure out what’s going on. Iteratively, I get a slightly clearer view, and discard a large number of the books as inappropriate. I will generally find a book in the first round of this and go off and read part of it. In this case it was Jean René Barra’s Mathematical Basis of Statistics, a lovely, Bourbakist book.

In the next iteration, I found Jack Carl Kiefer’s Introduction to Statistical Inference, which has a beautifully clear view of the mathematical structure of inference and how the various schemes such as Bayesian statistics fit into it.

The next day I was in the subbasement again — that’s where the math books are here — and this time unearthed John Tukey’s Data Analysis and Regression. Tukey steps back even further and spends a lot of time addressing bad stuff happening in the tails of distributions, situations when your data or techniques aren’t up to inference, robustness, and all the unpleasantness that reality can dish up.

And in parallel, I’ve been charmed by Edward Tufte’s inspiring The Visual Display of Quantitative Information. Don’t read it if you don’t want to spend the rest of your life foaming at the mouth when people put up bar charts or pie charts, or trying to make your graphs meaningful.

And then I presented this paper, which gives a few basic rules on error bars, to my lab in journal club, and I desparately wish I hadn’t. In reaction, I now proclaim: you cannot understand statistics without mathematical sophistication, and I’m not willing to try again.

So I’ll post gems as I find them, but I’m not going to write for nonmathematicians. Remember, there is no such thing as nonmathematical science.

madhadron :: May.05.2007 :: mathematics :: 2 Comments »

A friend who took his degree in mechanical engineering recently asked me for a plan of study to solidify his mathematical knowledge. He is already competent at multiplying matrices, integrating, and churning out solutions to the common families of linear differential equations. At the same time, he has no need of Bourbaki. Here is roughly the curriculum I outlined.

Start from calculus again. The introductory calculus book has one purpose: to get the student ready to work through baby Rudin. My recommendation is Spivak’s Calculus. Michael Spivak is one of the great mathematics writers of our day, and his book is designed to fill exactly this niche.

Once you have finished Spivak, it’s time to tackle Rudin’s Principles of Mathematical Analysis (a.k.a. baby Rudin for its small size). Rudin’s book is not fun, but you don’t need to remember everything in it after the fact. You do have to get comfortable with set theory, topology of the real line, sequences and series, and how to use these tools to construct proofs in analysis. Later this book is an invaluable reference. You might also put off the study of functions of several variables and integration of differential forms to Spivak’s Calculus on Manifolds below.

Linear algebra is next. Axler’s Linear Algebra Done Right is probably the place to go for this. The important thing is to learn the structure of linear maps. Matrices are just a computational tool — albeit a vital one — for dealing with such maps.

With analysis of one real variable and some linear algebra behind us, multivariable calculus is next. Spivak’s Calculus on Manifolds is probably the nicest book on this that I know. It’s also very short.

Complex analysis is the other important field at this stage, and it can be done before, in parallel with, or after multivariable calculus. Tristan Needham’s Visual Complex Analysis is an extraordinary exposition. He has managed to draw pictures of expansive parts of the field. The detailed analytical machinery tends to get short-changed, however, so I suggest combining it with a very terse complex analysis book of a more traditional character. You might look at the recommendations of Stephen Greenfield, though this is for graduate complex analysis.

Now we come to ordinary differential equations. I find V.I. Arnol’d’s Ordinary Differential Equations charming. You’ll need a couple things for this. First, everyone needs some group theory. Humphrey’s A Course in Group Theory is a fast paced exposition with a single goal: the classification of groups of order less than 32. It’s very clear. Then you’ll need differential geometry, and you could hardly do better than the first one or two volumes of Spivak’s five volume Comprehensive Introduction to Differential Geometry. He’ll also give you the material on Lie groups.

Unfortunately, from here I haven’t really explored the books, and what I know I learned directly from people.  But by this point, the student’s ready to be out on his own.

madhadron :: Mar.06.2007 :: mathematics, teaching :: 2 Comments »