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If I want to continue mathematics into a PhD program, would doing a bachelor's and master's combined program be a good idea? Would I be required to retake some graduate courses in the PhD program? This would add only one year to my degree from my current uni as opposed to the two from doing a BS and MS separately.

A bit of background information: I am a dual enrollment US student who has completed two years worth of college courses, and so I'll be entering as a freshman with junior standing taking upper division undergrad math courses. Thus, a standalone BS would take me only two years, leaving me to graduate at age 20. Is this too young to start applying to grad school?

Also, I plan to do an undergrad thesis; is one year of familiarizing myself with the faculty too little to choose an adviser? I'm not even sure which math I would want to write about. Will three semesters of intro analysis and intro algebra grad courses be enough to decide on the thesis for my fourth semester?

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    Mathematics is HUGE. You cannot call yourself a mathematician if you have not mastered (i) complex analysis and (ii) abstract algebra. Before then, there is no point in trying to decide what to do next, as most other topics will have bits and pieces of these two. If all you have had is calculus (without proofs, in the worst US route mem tradition), then you have a long road ahead of you. (Disclaimer: I had my abstract algebra and complex analysis in high school in Russia, which was enough to impress my measure theory teacher in my Ph.D. program by reproducing a proof from my 10th grade class.) – StasK May 31 '13 at 21:46
  • These classes are required in my curriculum, so I will fortunately be taking these soon. Do you consider complex analysis a prerequisite for real analysis, or vice versa, or a corequisite? BTW, I've been studying more advanced topics on my own since summer break started because I already know that calculus 1-3 is nothing compared to higher level mathematics. – anon May 31 '13 at 21:56
  • Real analysis first seems like a more natural procession, but this may depend on how deeply your profs will go into their particular versions of these two major topics. Ask around -- come to the office hours of the profs offering these courses; profs LOVE curious minds (I used to :) )... especially given how few American students are really interested in math. – StasK May 31 '13 at 22:00
  • Well that's good for my schedule since I plan to take real analysis first. And I will make sure to get involved with the professors after class hours, considering they've already dealt with what I'm going through! Thanks for the advice. – anon May 31 '13 at 22:26
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    @Jack, depends on how you define it -- I keep forgetting that say French theorems about extrema of differentiable functions are a part of real analysis in the U.S., not of calculus. (I had them as a part of a proper calculus sequence, Russian way: notion of the real line - sequences - infinite sums - single variable calculus with epsilons and deltas that relied on real analysis and sequences in say proving Bolzano-Weierstrass. American sequence, Stewart-style, makes little sense to me.) But my main point was about other areas having bits and pieces of these foundation topics in other areas. – StasK Apr 4 '15 at 12:48
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I finished my undergraduate degree at 20, and I know a number of others who have finished undergraduate considerably younger than I did. So, no, 20/21 is definitely not too young to apply for and begin a graduate degree program.

Since you list yourself as a US college student, here are a few questions to ask yourself before embarking on a dual-degree program:

  • Who is going to pay for the MS part of the program? You may not want to have to pay out of pocket the extra costs of the master's, when it would normally be covered if you picked it up "on the way" in the PhD program.

  • Are you happy with your choices for a research advisor for your master's thesis? (No point in being miserable by working for the wrong advisor—even if it is a semester or two, a miserable time is not what you want.

  • Where do graduates of your school's master's program end up for PhD programs? Are these schools you'd want to go to? (You can ask the same question of the bachelor's program, of course.)

  • Who will write your letters of recommendation? Do you have people who know your work beyond the classroom well enough to recommend you for external graduate programs? If not, this may vote in favor of staying for the master's program.

  • Do you have to make a decision about which program to do before you apply for external programs? No point in making a tough choice if you can keep it as a "fallback" option.

As for topics and coursework, I will leave those answers to the mathematicians.

  • Thanks for the input! I do hope to get to know higher level math better so I can decide what my thesis, undergrad or graduate, should be on, and choose my adviser according to their research. Also, do you know if there's a site for checking where people go from undergrad to get their PhD? I'd like to see where students from my college went to, but it seems that that information is not present on the school website. – Tii Jun 2 '13 at 0:21
  • I don't know of a website that does this—however, your department's undergraduate advisor may have this information. – aeismail Jun 2 '13 at 6:24
  • Alright, I'll go ask him about this-- he did mention having the statistics of all the students! Thanks again! – Tii Jun 2 '13 at 11:21
  • @Tii, there are people I know who get bachelor's at 18 or 19. Just saying. – Jack Bauer Apr 10 '15 at 10:35

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