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I'm considering graduating a year early and I'm having a bit of a hard time making the decision. I'm going to be as specific as possible so that I can get the best advice possible, so sorry if this becomes a bit lengthy.

I'm going into my third year as an undergraduate, majoring in mathematics. Here is a list of courses I have taken:

Calculus III, Intro to Linear Algebra, Differential Equations, Modern Algebra I, Advanced Linear Algebra, Modern Algebra II, Discrete Math, General Topology, Real Analysis I, Real Analysis II, Algebraic Topology, Complex Variables, Abstract Algebra I (a graduate course in Group Theory).

I am currently participating in an REU, and I am attending Math in Moscow in the Fall, where I will (most likely) be taking Algebraic Number Theory, Commutative & Homological Algebra, Riemann Surfaces, and possibly one other course.

I've done a fair bit of reading outside my courses as well: Group Representation Theory, some classical Algebraic Geometry (Fulton's Algebraic Curves, so I haven't been exposed to schemes yet), and Category Theory just to name a few.

Here are the upsides I see to graduating early:

  • opportunity to take more advanced coursework earlier (my university doesn't have an overly active graduate department, and it's likely that most of my learning would come from independent reading courses if I were to stay)
  • being in an environment with stronger students will probably force me to develop better as a student

and here are upsides I see to taking a fourth year:

  • more time to figure out what subject I want to pursue research in
  • opportunity to have more coursework under my belt before I apply
  • it might be difficult to apply from Moscow, so I wouldn't have to worry about that

I'd like to hear other people's opinions. I've of course asked my professors first. One is adamant that I should graduate early, he's actually the one who suggested it to me. Another says he thinks I'll be fine either way, it's just a matter of if I want that extra year to prepare.

So according to them, I should be okay. But I can't get the idea out of my mind that I might not get into as good of a graduate program if I graduate early, so I wanted to hear other opinions. Am I just being paranoid?

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    I suspect things are likely to turn out very well either way. One thing you might consider doing is applying only to one or two "dream" schools, and graduating early if either of them accepts you. Some additional questions you might ask yourself: (1) Is undergraduate tuition a serious burden? (2) Are there any non-math courses you'd like to take while you're still an undergraduate? (3) What do your REU instructors think? (I'd ask near the end of the program.) (4) Are your writing and public speaking skills well developed? These will eventually matter a lot. (5) What, in your gut, do you want? – Anonymous Jun 14 '16 at 22:58
  • @Anonymous thanks for the response. Tuition isn't an issue, and there aren't many non-math courses I'm keen on taking. I'll be sure to talk to my REU advisor towards the end of the summer. I really like your idea of applying to a few programs and seeing where I get into before I officially apply to graduate, I'll probably do that. – Alex Mathers Jun 14 '16 at 23:37
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    @Anonymous Can you please turn your comment into an answer so that I can vote for it? – jakebeal Jun 15 '16 at 1:52
  • @Anonymous: I'd like to vote for it also, as that was great advice. – Dave L Renfro Jun 15 '16 at 14:20
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You are obviously in a good position either way, but as you suggested, graduating early appears to be your "practical" decision.

If you are considering a doctorate or masters (it sounds as though you are), graduating will allow you to get a head start on it.

However, if you you truly don't know what what graduate degree to pick, and don't decide to go with that, consider doing an independent study with a favored professor or doing some tutoring at your local university, as it would not only contribute to your resume when applying for a masters program, but also give you opportunities to dabble in subjects you find interesting, and ultimately help you decide which field you'd like to master in.

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