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I know this is not the forum for this particular question, but the majority of users are immersed in the environment. I want to study differential geometry, but I need to know what courses would help guide me in that direction. I enjoy and/or also want to study the following as well: abstract algebra, real analysis, partial differential equations, complex analysis, functional analysis, and topology. Which of these topics should I study when I start next year?


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closed as off topic by Ryan Budney, Andrés E. Caicedo, Felipe Voloch, Andy Putman, Dan Petersen Jun 18 '11 at 19:46

Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here.If this question can be reworded to fit the rules in the help center, please edit the question.

Talk to your advisor. – J.C. Ottem Jun 17 '11 at 19:32
I want to compare what others are doing from different campuses, as oppose to one person's word. Any input is greatly appreciated. – Steppenwolf Jun 17 '11 at 19:45
Steppenwolf, it is hard in these situations to give advice if one does not know your particular circumstances, aptitudes, background and so on. This is why I (like J. C. Ottem) think you should talk to someone in person. It would be easy for many of us to make suggestions but hard for us to feel confident they will actually do you good. – Yemon Choi Jun 17 '11 at 22:52
To build in Yemon's point: we don't know what graduate school you'll be attending, how its program is structured, and what classes will be available. You should talk to someone who knows that stuff. – Ben Webster Jun 18 '11 at 5:49
Steppenwolf, this question is being closed because MO isn't the appropriate forum for it, and not because it isn't important. I am sure many in the wider community of mathematicians do want to help, but MO is not set-up for discussion. I hope you'll continue to remain an active member, and answer/post research-level math questions for many years to come. Good luck! – Nilima Nigam Jun 18 '11 at 19:53
up vote 20 down vote accepted

The sociological and metamathematical aspects of this question are too often overlooked, I think. First, undergrad or grad students' discussions with their peers too often subtly veers into a "Lord of the Flies" scenario. Second, many "advisors" (whether undergrad or grad), have some weaknesses in communications skills and in perception of others' non-verbal expression (e.g., affect). Nevertheless, yes, one should talk to faculty quite a lot (even if taking remarks with a grain of salt).

Now some objections: the labels of "subjects" or "specialties", while seemingly sanctioned or even mandated by the AMS subject classification, by faculty "research descriptions", and grad students' desires to taxonify ambient activity, are innately misleading. There are no clear separators... except those artificially imposed. True, the "requirements" have such labels, and "everyone" speaks in terms of them, and... yes... one can live one's whole professional life speaking in those terms, ... but this partitioning is fundamentally invidious.

The next objection is that it is usually very difficulty to understand the significance of things until one sees how they're used "in the sequel". Thus, a misguided fixation on "mastery" at an entry level really is misguided, in that one is doing exercises without a notion of real-life activity. The utility of things is not well-illustrated by contrived (a.k.a., "textbook") exercises.

These remarks are all cliches, but perhaps bear repeating...

Edits: Seeing the responses, I'd like to add clarifications. First, one should not depend on coursework for learning mathematics, especially not for seeing how it is done in real life. One must learn more-and-different things than what the traditional curriculum promotes, no matter which courses one signs up for. One special corruption is the usual convention of assigning piles of weekly homework, exams, grades... leaving people little time or energy to think critically about anything, and confusing compliance with scholarship. (Observe: in mathematics, apparently one is not permitted to question the goodness of course content, insofar as grading systems reward obedient technical responses rather than critiques.)

Far more important is awareness of things, of their utility, of their interactions with other things. Earliest-possible awareness of as many ideas as possible is highly desirable, whether or not piles of exercises are completed.

Thus, it is desirable to "look at everything", and obviously this can't happen via coursework. It is desirable to witness the actual practice of mathematics, thereby to be aware how different doing mathematics is from doing homework or exams or contest problems. Seminars sometimes represent this, although often they amount to reports or job talks. Regular conversations with faculty about mathematics, not about coursework, surely cultivates a more useful outlook than any amount of coursework.

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I get what your saying and I have spoken with faculty members. I'm also assuming you're a faculty member based off your profile information. I know by talking to a variety of people, I will get varying perspectives but I believe that is the part. Where would you recommend starting, or going from if you were a student again? Thanks. – Steppenwolf Jun 18 '11 at 3:23
@Steppenwolf: Please read the comment to my original response below. – GH from MO Jun 18 '11 at 8:28
Thanks Mr. Garrett. I think that is the perfect answer. I'll just study many topics that happen catch my interest and go from there. I will definetly keep your advice in mind, especially since my upper level courses were taught from a seminar perspective. Thanks again. – Steppenwolf Jun 18 '11 at 20:07

First study abstract algebra, real analysis and topology. Then study complex analysis, functional analysis and partial differential equations.

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+1 this seems like a short but clear and good answer to me. – user9072 Jun 18 '11 at 7:42
@unkown: Thanks. It seems short and clear answers are not in fashion, my response got "-1" votes so far. What I really meant is that: all 6 subjects should be studied as they are all basic, but 3 of them should clearly precede the other 3. – GH from MO Jun 18 '11 at 8:26
I am pleased to see that my response has now a positive balance. – GH from MO Jun 18 '11 at 20:16
I think people (by which I mean me) downvoted it because it didn't seem to be really addressing the OP's question, and didn't contain information that couldn't be learned from looking at a course catalogue. It also has what I would call a false precision: the fields listed above are very broad, so whether your answer is right depends a lot on what those words mean to the OP. – Ben Webster Jun 19 '11 at 23:07
@Ben, fair enough. Hopefully my response was of some value to others. I answered the question the way I understood it, and indeed I did not think of the course catalogue which might contain similar information (namely prerequisites). – GH from MO Jun 20 '11 at 8:51

(1) Many graduate programs have a relatively fixed curriculum for first year students. Some require courses to prepare you to pass exams, others require courses for exams you don't do well enough on when you arrive. So you may not have complete freedom when you arrive, and only an advisor at the school you are going to can help you with that.

(2) If you are going into a PhD program, you should keep in mind that you will need to transition relatively quickly into a specialization (within a couple years at the longest). You have to write a dissertation for a PhD, and that means finding a thesis advisor and taking specialized courses to prepare. How quickly this transition happens depends, again, on what school you are going to.

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To respond to JC Ottem's response that got 11 upvotes in the question's comments section:

At most universities these days you do not have an adviser when still in class work. This comes from someone who is in the midst of it and has many friends in the midst of it. So, that's a harsh one line statement you made to someone looking for valuable advice.

Edit: at most universities ... in the united states

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Regarding your "at most universities" -- that may be true in the States but it's not true globally. – Ryan Budney Jun 18 '11 at 19:41
@Ryan good point, my apologies on the lack of clarification; thank you for pointing this out. – Andrew Jun 18 '11 at 19:53
Andrew- In my experience, typically they do have a faculty member who is supposed to give them advice on things like classes if the student wants to ask for it (for example, at Oregon, there is a "Graduate Advising Committee"); when I was a graduate student at Berkeley, there was a faculty member who had to sign off on my course schedule every semester who I was assigned on my arrival (and was not my thesis advisor). So "Ask your advisor" may been the wrong formulation; the OP is in a very deeply flawed graduate program if there's no faculty member s/he can go to with questions like this. – Ben Webster Jun 18 '11 at 21:46
@Ben Yes, in retrospect my issue was more with formulation, as you describe above. – Andrew Jun 19 '11 at 0:57

One of the reasons to study many courses is to gain sufficient breadth in ones education, and to have handy the information when needed. (Despite the Internet, one's brain is still handier to have, and one's perspective is important in considering applicability of knowledge. Internet resources will never be able, in my opinion, to reduce the role of perspective in determining what information is applicable and how.) Another reason is to find out what one likes and then do as much of that as compatible with one's other goals in life. So take the standard route and vary it at your own discretion; advisors and mentors may be helpful, but they need to know more of you; advice gotten from the Internet is rarely worth more than it costs to get it.

All the courses you have and more are recommended, but the pace and organization is something you and someone familiar with you and studying should work out. Also, if you have ideas on how to go about something, tell someone. One of the biggest faults I had as a graduate student was keeping too much to myself because I thought I had to be original in most everything I did. The reality of graduate school is that your work will build upon others and that one or two ideas on how to do something new, plus a lot of academic and other necessary grunt work, is what will help you get your degree or get your ideas properly recognized.

Good Luck. (Yes, sometimes luck is useful in getting a degree.)

Gerhard "Ask Me About System Design" Paseman, 2011.06.17

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One way to make the point in the first paragraph: the Internet can help you search for things, but (as of yet) it is not so good at telling you what to search for. – Qiaochu Yuan Jun 17 '11 at 23:23

Steppenwolf, the strictly pragmatic advice would be:

1) Identify the graduate programs in differential geometry you'd like to be a part of, and look at their first year coursework. Many programs have written qualifying or comprehensive examinations, so the coursework may be structured around it. I would imagine analysis, algebra and topology would be key ingredients in most of these program.

2) Identify some of the people you'd like to work with, and maybe ask them in person? This would have the side-effect of learning about these people as potential supervisors.

Best wishes for your graduate career! I hope you find it a pleasurable (even if unpredictable) journey.

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This answer just seems to underline how unclear the question really was. You seem to be writing as though the student were trying to choose a graduate school in Canada, when my reading was that they had already chosen one in the US, and were trying to choose their classes. Of course, I think I interpreted things correctly, but the OP should have said a lot more if they wanted useful advice. – Ben Webster Jun 19 '11 at 6:08
Agreed entirely. Even 'next year' is ambiguous, since it could be the next academic ('11) or next calendar year ('12). That is why MO is not the correct place for this question. For advice to be truly useful in this instance, the possibility of iteratively asking for pertinent information must exist. That becomes a discussion. – Nilima Nigam Jun 19 '11 at 14:59

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