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I am an undergraduate at a large public research university in the US, finishing my second year. I have done some theorycrafting around my graduation course requirements, and I have realized that it is possible, with some work (and generous CLEP testing), to graduate with a double major, Math BS + Physics BA, by next May. I would be meeting the bare-minimum requirements for both majors (in particular, for math, I will have only taken the 4-semester calculus series, lin. algebra, and the 2-semester abstract algebra series; plus some extraneous ones e.g. game theory).

If I were to stay the extra year, I would comfortably qualify for a double Math BS + Physics BS, and I would have 3-4 more math classes under my belt (perhaps the 2-semester topology series, graph theory, advanced abstract algebra, real analysis, etc). (edit: this option, Physics BS rather than BA, stresses a more thorough physics courseload; 15 hours of upper level physics classes are required for BS over BA)

edit after several answers: Another 4-year option is to get Physics BA + Math BS, leaving my course load open to taking 3-4 math classes per semester. This, hypothetically, could put my post-lin-algebra courses at 2 sem. abstract algebra, 2 sem. topology, 2 sem. intro analysis, graduate level courses of the above, intro complex analysis, intro graph theory, etc

My intent, if I were to graduate next year, is probably to take a year off from academics, and then apply for graduate/PhD programs in math.

Will it be an issue in being considered for these programs, that I have only taken a meager selection of the course offerings in my undergraduate studies? And might it be offset by the fact that I graduated in three years with a double major?

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The only person who will be impressed by your finishing in three years is you. School is not a race. –  JeffE Apr 12 at 22:47
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I'm confused. You want to get your undergrad finished as quickly as possible, spending half of even that short time doing something other than maths. You then want to take a year out, again not doing maths. One minute, you're saying that you want to spend four years of doing as little maths as you can get away with; the next minute, you're talking about a PhD in the subject. That seems inconsistent. –  David Richerby Apr 13 at 12:51
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I have to ask: if math is "the only subject you enjoy" (which I hope is an exaggeration; there are lots of other interesting things in the world), then why are you so set on getting that physics major at all? As far as grad school and a career in math, it may hurt more than helping. Of course, if there are a couple specific courses that really interest you, you could certainly take them. –  Nate Eldredge Apr 14 at 0:01
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It also might be worth exploring why you are so anxious to get out of college... –  Nate Eldredge Apr 14 at 0:03
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@NateEldredge The physics major's already complete, save one class. Might as well finish it. I came in as a physics declared major, not math. –  Justaskin_ Apr 14 at 2:23
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4 Answers 4

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Your question makes me a little sad. You are obviously a bright and ambitious student, and you have spent almost two years at a US public research university. You are trying to "theorycraft" your future career, which is good. But somehow no one has managed to communicate to you the basic goals of an undergraduate math major and the requirements for graduate study: how did that happen?

To once again answer the title question (the other two answers are completely on point): "Does rigor/thoroughness of undergraduate program matter (for graduate/Phd applications)?" The answer is yes, of course. Not only does it matter, but together with indications of your success in these courses -- course grades together with recommendation letters from your instructors -- it is what matters the most. Your question first proposed graduating early with a double major in physics and math, with coursework that to any math PhD admissions committee member looks extremely minimal. This is a terrible plan for getting into a PhD program in mathematics (not just a top one, but any one that I know).

Let me be clear: I am not scolding you. I am lamenting the total lack of advice you have somehow been subjected to. You are not the first bright, ambitious undergraduate who has asked a question on this site which evinces the misconception that the best students get through undergraduate degrees in as few years as possible. This is quite the opposite of the truth: does anyone know where these misconceptions are coming from?!?

Here is my opinion (as someone who served on the graduate committee of a top 50 PhD program in mathematics for four years): unless you want to work cross-disciplinarily in math and physics, the second undergraduate major in physics is not directly helping you get into a PhD program in mathematics. On the other hand, it seems to be stopping you from taking the deeper and more advanced course offerings that are essential for you to even get consideration at a good (or even decent) PhD program in mathematics. If your postgraduate plans include a career in which knowing a little bit about several different things will be beneficial -- and I can imagine that many careers in business and industry are like this -- then a double major could be desirable. However, for a future PhD student, all else being equal a double major is time taken away from the one thing you're supposed to be devoting much more time and energy to learning.

In your case, you absolutely need the fourth year of your undergraduate program in order to be considered for a top program. Unless you absolutely load yourself up with challenging and graduate-level math courses in that last year (which may not be the best idea for other reasons, e.g. it works against the level of your success in these courses), I would in fact be thinking about what to do after you graduate in four years in order to make your grad school application more competitive. You say that you want to take a year off before going back to grad school. If you still want to do that after a four year degree, then thinking about how to get some additional mathematical training in during that year off might be a good idea.

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Unfortunately, I've already completed all the course requirements (except one) for a physics BA, so at this point I might as well complete the degree instead of leaving 20+ credit hours wasted. The physics double major is not really an "option", it's a decision already made. I came in as a physics major, not math. –  Justaskin_ Apr 14 at 2:18
    
The people who have (mis)represented math undergraduate study and beyond, that would be my generic arts&sciences adviser, and my adviser in the physics department (physics having been my major for 3 of my 4 semesters). The physics adviser's attitude toward math was one of my first indications that I might not want to continue physics; one quote I specifically remember from my advising sessions, "A physicist isn't concerned about why there aren't solutions to fifth degree polynomial equations." What? Why not? –  Justaskin_ Apr 14 at 3:03
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This would be a great question to discuss with your advisor, or another faculty member at your school whose opinion you respect.

My opinion, for what it's worth, is that the "bare-minimum" 3-year program you describe would be woefully inadequate to prepare you for graduate study in mathematics, and I think most admissions committees would agree. The fact that you did it in 3 years with a double major won't mitigate that. A year of analysis is pretty essential, some topology would be very helpful, and taking a significant number of upper-division math electives will be a great way to build what people call "mathematical maturity".

Your 4-year program sounds better, but honestly still a little on the light side. Keep in mind that many applicants to top grad programs will have taken practically every undergraduate course their department offers, and maybe a couple of graduate courses besides, in addition to often having undergraduate research experience. You also want to make sure you get to know (and impress) your mathematics professors well enough that they can write you strong letters of recommendation, and taking more math courses is the biggest part of that.

The physics degree on your diploma will be nice, especially if you are looking to do a PhD in mathematical physics, but it may not go a long way to making up for weakness on the math side.

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Another course of action for the 4-year is to just stay Physics BA instead of BS (this cuts off 5+ strenuous classes and labs) and focus more on fleshing out the math BS. This may be the best choice for me. –  Justaskin_ Apr 12 at 16:19
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While Nate's answer mostly covers it, let me be a little more concrete: I just went into the files for graduate admissions at my own institution (the University of Virginia), and looked at the double majors we had admitted or wait listed. I can't give details obviously, but I can say they are all way ahead (curricularly) of where you propose being at the end of your 3 year degree, and probably of the 4 year degree you suggest (6 classes beyond linear algebra is actually not very many). I don't think there's any way that the 3 year degree you suggest could get you admitted into a top 50ish school (in USNW or NRC rankings) without some sort of exceptional factor, and if you have 6 post-linear-algebra classes, your background will still be a big disadvantage, unless you are looking specifically at applied math programs where your physics background will actually be taken into account.

To answer your general question: yes, rigor/thoroughness of undergraduate program makes an enormous difference. Graduate programs are under pressure from university administrations to graduate students in a timely manner, and generally don't want to have students with weak backgrounds who will hang around in the program for a long time (or flunk out). Most schools have a set of exams you need to pass before moving on to your thesis, and (just as an example), at UVA, you must pass a topology or analysis exam within 2.5 years of starting the program. Thus, admitting someone with no topology or analysis background would just be setting them up to fail those exams.

While obviously, there's some sort of nebulous notion of talent that schools put a higher premium on (so you also want strong grades, and strong recommendations), background is relatively easy to measure, and is a very important factor in admissions decisions.

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Thanks for your answer. I've edited my question to note another option which allows me to focus on, as you put it, "post-linear-algebra courses". –  Justaskin_ Apr 13 at 19:32
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Seconding the other good answers, I would reiterate that, unless you are doing massive amounts of documentable self-study (which is entirely possible modulo the documentability), the senior-level undergrad or beginning grad-level courses you take substantially affect the appearance of your grad school application in at least two ways. First, your documented exposure to something approaching real mathematics. Second, the letters of recommendation from the faculty who taught those courses, testifying to your (presumably...) excellence in beginning-serious mathematics. Letters of recommendation are a very serious component of grad school applications, and if people can only refer to things scarcely beyond linear algebra, it won't help you.

Indeed, as @Pete L. Clark observed, you are being dis-served by lack of advising about the goals of your undergrad program, and about preparation for grad school. I realize that there are many ways for young people to disconnect from useful-but-official information, and many ways for institutions to fail to communicate effectively, but, either way, you're missing a bunch of critical info, which a site such as this can only nudge you toward.

One more time: the "minimalist" version of a math undergrad degree will doom your grad school applications. Finishing in three years versus four doesn't matter at all.

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