Foundations book using category theory for student embarking on PhD in mathematical biology?

Question

I'm about to embark on a PhD in mathematical biology. My major is in computer science.

I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research problems. My plan is, initially, to go through Linear Algebra Done Right by Axler, and Principles of Mathematical Analysis by Rudin as a refresher.

I was considering to go through a foundations book like Naive Set Theory by Halmos beforehand. But perhaps a category theory-based approach could be more enriching, as it could help me to see connections between many areas and concepts in math. Is this a good idea?

Sets for Mathematics and Conceptual Mathematics, both by Lawvere seem to be popular choices. The former seems to be a nice construction of set theory using categories instead of ZFC, but not much more than that. The latter seems to spend more time addressing the connection between categories and different branches of math, like linear maps. Any suggestions?

Answer 1

From my very limited knowledge of mathematical biology I was under the impression that the more relevant areas of mathematics will be the "applied" ones, as such I don't think that reviewing the abstract areas of mathematics will be very helpful too you. My other thought is that without doing a maths degree it will be very difficult to cover much ground in areas such as Category theory (which I only studied having completed an undergraduate degree in pure mathematics) especially if you haven't seen a lot of very algebraic mathematics (rings, groups and modules etc.).

Areas I think are often applied to parts of mathematical biology are things like dynamical systems and stochastic differential equations. As such, Rudin would be an excellent starting point and then maybe something like 'probability with martingales'. Axler's linear algebra book is fantastic(although not applied). I would stay away from pure set theory simply because most mathematicians who don't study foundations don't need it on a day to day basis.

Basically I think that acquiring a more rigorous understanding of mathematics is a fantastic idea, but the best way to prepare for your research would probably be to find out exactly which areas of mathematics you will be using (broadly speaking) and then study that specifically.

Answer 2

As Artem helpfully suggests in a comment to the question, you might want to try David Spivak's Category Theory for Scientists (Old Version). See also the MIT OpenCourseware page and John Baez's blog entry.

You might also be interested in Robert Ghrist's Elementary Applied Topology, noting that Chapter 10 takes a category-theoretic perspective and that computational geometry and topology are increasingly being applied to biology (see, for instance, the work of Gunnar Carlsson, Herbert Edelsbrunner, Vidit Nanda and others).

I hope this complements Edward Hart's answer, which is probably more relevant if you plan to focus on dynamical systems and the like, although even then you may benefit from studying abstract mathematics and subjects such as topology (including computational homology, which can be used to help analyze dynamical systems). I agree that his advice to "find out exactly which areas of mathematics you will be using (broadly speaking) and then study that specifically" is eminently sensible. I would just add that studying category theory could help in developing your conceptual thinking skills, regardless of the specific areas in which you become involved, and increasingly it has applications beyond foundational matters.