Best way to prepare given my background?

[Mathemagician]

Hi all,

I graduated last year from a small liberal arts college with a double major in math and economics. After working for a year I've decided I want to prepare an application to a math PhD program. I purchased a few textbooks in topics I hadn't covered (complex analysis and differential equations are the big ones I haven't seen, and I also purchased a grad school text on analysis) and have been reading through them at a casual pace, but after a glance at this forum it's obvious I need to make a more deliberate and focused effort.

I've taken courses in topology (with some algebraic topology), advanced abstract algebra (including Galois theory), advanced set theory/foundations (including axiom of choice and equivalents, Godel theorems, ultrafilters), linear algebra, and real analysis, as well as some more basic courses (calculus, cryptology, etc.) with an A- in Calc 3 and an A in everything else. I performed summer research in voting theory which progressed into an honors thesis; my advisor is currently publishing a paper with me as one of the co-authors. I am extremely close with the faculty and should have particularly strong letters of recommendation.

Unfortunately a hell of a lot of the problems I've seen here require techniques I haven't been trained in, so I'm a little worried and fear that reading up on subject after subject that I haven't seen may not be the best way to prepare. In addition, I feel like I may need to invest at least some time reviewing previous coursework. So I'm feeling overwhelmed and unsure how to proceed.

Any advice would be greatly appreciated!

[owlpride]

Are you studying for the Math Subject GRE or in preparation for graduate school?

If your primary objective is the GRE, studying complex analysis or differential equations is a very inefficient use of your time. There will most likely only be a single complex analysis question on your exam, which you might just as well skip. As for differential equations, on the three exams I took (two released past exams and my official one), I only needed to be able to solve two basic types of ODEs: (systems of) linear equations and equations of the form f'(x) + g(x) * f(x) = h(x). You should be fine if you memorize the solutions to those two, no need to work through an entire diff eq textbook. (I have never taken a differential equations course either!)

I don't know how much discrete math you have seen. Combinatorics and elementary number theory are probably the most frequently tested topics after calculus and linear algebra. They are worth studying if you haven't been exposed to them before. (Luckily they are also very easy to learn.)

Other than that I am positive that you have seen all of the relevant math in your previous coursework. Most if not all of the GRE questions can be solved with elementary techniques from each field of math, but the questions might be stated in a non-standard form that makes it not obvious how to approach them. In that sense the Math GRE is really more about being clever than knowing a lot of math. The test seems to appeal to students who are also interested in problem solving competitions.

[MathMudblood]

Hey Owlpride,

What are good books to get to help study number theory and combinatorics for the GRE math subject test?

Cheers,

- Mudblood