Going to graduate school for mathematics next year, need some advice - MathOverflow [closed]

Going to graduate school for mathematics next year, need some advice

2011-06-17T19:20:43Z

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?

Thanks

Answer by GH for Going to graduate school for mathematics next year, need some advice

2011-06-17T19:31:13Z

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

Answer by paul garrett for Going to graduate school for mathematics next year, need some advice

2011-06-17T22:59:14Z

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.

Answer by Carl Mummert for Going to graduate school for mathematics next year, need some advice

2011-06-17T23:09:57Z

(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.

Answer by Gerhard Paseman for Going to graduate school for mathematics next year, need some advice

2011-06-17T23:15:18Z

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

Answer by Andrew for Going to graduate school for mathematics next year, need some advice

2011-06-18T18:53:38Z

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

Answer by Nilima Nigam for Going to graduate school for mathematics next year, need some advice

2011-06-18T19:45:39Z

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.