How many math classes should I take a year and of what sort?

Question

I'm going to be a junior in college next year and I've only taken abstract algebra. I plan to take three math classes next year - analysis, topology, and logic (plus a class on "stochastic processes"). I am afraid that this will kill me (it might not), but I am also afraid that if I don't take a lot of math I won't get into/be prepared for a good grad school. (My professor said his first year generals exam involved representation theory :| how typical is this?) How many math classes do you recommend I take a year?

Also, for senior year I get to choose between computability theory, graduate algebra, graduate analysis, combinatorial analysis, algebraic geometry, probability (with measure theory), ordinary and partial differential equations, dynamical systems, algebraic and differential topology, algebraic number theory, and introduction to applied mathematics. Which of these should I take to be best prepared for grad school? I am required to take combinatorial analysis. Also, I'm not the best at algebra but half the classes require it and I don't think I'll be good at analysis either if I have to take three classes at once.

Thanks :)

Oh, and I might want to go to grad school for political science or biology so maybe I shouldn't take too many classes in math, but I'm kind of afraid that by dividing my efforts I'll be mediocre at everything.

Edit: Most of the courses I mentioned are year-round, so I actually can't take as many of them as I would if they were one term each.

Answer 1

Your best bet is to talk to people around you: your major advisor, faculty in the department, math graduate students, fellow students looking to go to graduate school. Ask what they did for preparation, and see what they recommend. Also, ask them why they chose graduate study – their perspective may be useful if you're on the fence about going.

Speaking from my experience as a recent college graduate:

The hardest classes in my major were the proof-based, two-semester sequences of the mathematics core, abstract algebra and real analysis. Most people I knew avoided taking both at the same time, because they took a lot of energy and thinking.

The other classes, mostly electives, were not as time-intensive. I wound up taking about 2–3 math classes out of 4-5 every semester after freshman year. The key for me was making sure I evened out one or two difficult classes with two or three easier classes.

To get a feel for what incoming PhD's were expected to know, the first year preliminary exam is a combination of calculus, abstract algebra, linear algebra, and real analysis, with a sprinkling of topology. Grad students take the prelim twice: at the beginning of the program to determine whether they should take the masters level or the PhD level versions of algebra/analysis/topology in the first year; and at the end of the first year to ensure that everyone has a base level of knowledge. I think they save advanced topics like representation theory for the second year oral examinations.

Undergraduates who are thinking of graduate school were highly encouraged to participate in an REU (Research Experience for Undergraduates), so that you know what you're getting into.

Answer 2

If you're going to grad school in math then I'd recommend taking at least four solid math courses per semester. In American schools there's a hell of a difference in the workloads going from undergrad to grad. It's best to get used to spending all your time in your area and getting used to the pace of grad school.

Regarding your second question about which courses to take, you need to ask yourself two things first:

• Do you find yourself attracted to a particular area of research?
• What are the focus areas of the departments you will be applying to?

When you know those then we can give you more specific advice on how to choose between those options you've listed above. In the meantime, Michael's advice on the prelims is excellent. Those prelims are usually created by the current (and past members) of the faculty. There's no better way to get a view on the department thinks is core than by looking those over.

A couple other notes:

• Grad school's going to prepare you for research.. NOT a job. -- Don't be misled into thinking you're gonna come out with a masters in pure math -- most departments are looking for phd candidates due to the time investments they make in their students.

• When in grad school you gotta learn how to zoom in on open problems that are tractable and in the domain of the faculty so they can guide you as your skills develop. Learn the core stuff (real and complex analysis, topology, geometry, and algebra) stone cold. That's what's going to make it easy for you to become self-sufficient and build out the skills you need as you move towards an open problem that will become your thesis.

• The nature of research in math is changing. It's becoming more experimental and computational due to the advent of killer computer algebra systems (e.g. SAGE, mathematica, Pari, Singular, etc..) Get good with one of them (and I mean REALLY good), learn how to code (Python is nice), and make sure you hook up with an advisor (and department) that embraces their use. It's where math is going and if you don't end up in math when it's all said and done -- we'll take you in finance. :)

Answer 3

Take as many as you want to, and as many as you feel you can.

I am going to be a little a blunt here: Why do you feel you want to go into graduate school for math when three courses over a whole year "might kill you?"

I don't mean to be discouraging, but if you feel negatively towards spending an enormous portion of your time doing mathematics, why would you want to continue on and spend 4-5 years of your life doing research?

Don't take mathematics courses out of fear of a poor graduate school. Take them because you are genuinely interested, and you would be thinking about it all the time anyway.

Sorry this isn't this advice you wanted,