I'm near the end of my pure math undergrad trying to decide what sort of math I'm interested in for graduate school. I've always thought the idea of mathematical biology was cool, but it seems like a lot of it is steeped in differential equations, which I'm really not too big on. I know there's some genetics research related to statistics, too, but I'm more into pure math. I guess the question is, is there any area of mathematical biology that doesn't rely heavily on differential equations or statistics (and if so, could you provide reference material)?
Misha Gromov's writings do not involve any differential equations.
You can also be interested in topological aspects of the study of DNA.
Here is another link that studies the so-called evolutionary algebras.
There are a lot of problems in mathematical genetics that involve dealing with various stochastic processes, but it seems that it is not what you are asking for.
All the stuff I've seen involves birth-and-death Markov Chains, especially SIS (susceptible-infected-susceptible) and SIR (susceptible-infected-recovered) models, it involves probability theory, asymptotic convergence and recurrent equations. The applications are vast - population biology, immunology, viruses. See Wierman, Marchette (2004) on applications to computer viruses and Nasell (1996,1999) on virus propagation.
Yes, I know about two such things:
DNA reading and computation (combinatorics on words, theory of languages).
Trying to explain the evolution in terms of mathematics (Darwin's theory is nice, but doesn't explain why 2 sexes exist. There're people claiming that 2 sexes means that you do somewhat "global optimization" while having no sexes would yield to "local optimization".)
Unfortunately, I cannot explain you this in much more details, since I know both only from more-or-less popular talks.
You ought to get a copy of THIS somehow. The one thing I remember discussing once that may not be represented very well is transform methods. Most of the body, the heart and lungs, are healthy when periodic, so ODE techniques come up automatically. It is different for the brain, much brain activity is aperiodic. The particular topic, though, was an experiment the guy was doing with brain-damaged patients who listened to bird songs. His one comment was that hearing was closest to a wavelet transform.