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I'm from the top math university in my country (it's not really world class but on average one guy from every other batch in my uni goes to Harvard/MIT/Princeton/UChicago/Stanford/UMichigan for PhD).

I'm in my second year and I'm broadly interested in number theory and geometric analysis, but there's no "good" faculty in number theory in my university (there's only one prof and he don't do any active research and don't have much reputation outside my country), so I'm almost on my own to study number theory.

It's pretty hard to learn everything by myself with simultaneously handling the coursework, so I'm wondering whether it's a good/sensible idea (wrt say grad school application where I need to compete with many people from top colleges in USA who are taking legit number theory courses by world class number theorists) to switch completely to geometric analysis and ditch number theory ? I think I can focus more on geometric analysis in that way too -- the only thing that is bugging me from switching completely is that I like number theory more than geometric analysis.

There were one guy from my college who went to Princeton for number theory who wrote a solo number theory paper as an undergraduate, but I'm nowhere close to being that smart. Also I can do reading courses/summer internship under some active number theorist researchers in my country, but the problem is that you need to know some nontrivial amount of number theory before you can do some internship under some reknowned number theorist !

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  • You seem to want inconsistent things. A top professor in a top university, who is therefore very busy and likely has many students, but who can give you a lot of support, which takes time and effort. It may not be a realistic expectation.
    – Buffy
    Mar 6 '20 at 23:11
  • @Buffy Thanks, but I'm not wanting a top prof in a top university (as I mentioned again, I'm a sophomore and I've no plan to change to another uni - so I'm stuck with the current prof if I want to do number theory)
    – noobie
    Mar 6 '20 at 23:14
  • 1) you can study number theory in grad school in the US without having learned much as an undergrad. 2) if there's no one to teach you number theory, have you considered transferring schools, or at least participating in special programs (summer schools, budapest semesters, ...)?
    – Kimball
    Mar 8 '20 at 19:59
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Number theory is perhaps the historically deepest, most studied area of mathematics. To even begin to understand research in most areas of number theory, you need two full years of graduate coursework over several subfields of mathematics.

What one can learn in an undergraduate course, or even a first graduate course, in number theory is so far from the frontiers of research that it makes absolutely no difference whether the person teaching is active in research or even has ever done research in number theory. You probably won't get into the 20th century in an undergraduate course, and not much beyond the 1940s in a first graduate course.

Except for a few exceptional students who are taking several graduate courses while still an undergraduate, no one is doing serious, mainstream number theory research as an undergraduate. Hence, it doesn't matter. You're not competing with people who have somehow learned lots of number theory already.

I should add that, in the job market for permanent positions in pure mathematics that support significant amounts of research, the sky has fallen. Only exceptional people are getting permanent research-oriented jobs now. Given that your job after your PhD (and your postdoc if you get and do one) probably won't involve doing research in either number theory or geometric analysis, you might as well study what you're interested in.

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  • Thanks a lot for your answer. Taking graduate courses is the norm rather than the exception in my uni... in fact the last year is solely for taking graduate courses. I won't most probably do a postdoc or stay in academia, but just curious, why what I do in my postdoc will not be related to my topics of interest ?
    – noobie
    Mar 7 '20 at 7:31
  • You parsed that differently from what I intended - I meant job after (your PhD and your postdoc ..), not (job after your PhD) and (your postdoc) Mar 7 '20 at 21:03
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It is true, as @Mr Cabbage observes, that there probably are economic realities to be considered. Unless you're independently wealthy, you'll eventually need to have a paycheck. And probably a reliable one. If you're "on your own", with no responsibilities to anyone else, you can obviously take more risks than otherwise.

So.

One point is that you have just one (immediately apparent) life, and if you are fond of number theory and don't pursue it, it'd be a pity. But, yes, risky, especially in your situation of not having senior people to give you good advice about it.

Also, in my experience in the U.S. at a "research univ", and having been involved with grad admissions for 35+ years, sensible people on admissions committees are well aware that people applying to grad school are very much in formative, transitional parts of their lives. What I myself look for is enthusiasm and self-starting, as much as anything. So, for example, the possible fact that your affection for number theory might compel you to pursue it despite lack of support in your immediate environment is a plus... while I'd know that you'd have some disadvantages in terms of advice and mentoring.

That is, to first-order approximation, do what you want to do. Second-order: try to make a living at it. :)

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I think you should do what your heart tells you to do. If you love number theory, do number theory. However, there are certain economic realities to take into consideration. Most people study with the intention of getting a job afterwards. Therefore, since you have a talent for mathematics, I would suggest that you become a statistician, since there are far more jobs in statistics than in any other area of mathematics. You can make a decent living as a statistician, and pursue number theory in your spare time. The fact that the distribution of prime numbers has properties akin to a probability distribution, in view of the prime number theorem, indicates that work as a statistician would not be inconsistent with doing number theory as well. Also, there are primality tests that involve probability arguments, such as the Miller-Rabin test. I would suggest statistics-cryptography-number theory is the way to go, in view of your talents.

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Is it reasonable to do [SUBJECT X] without active/good prof in uni

I will answer generally, since whether number theory or something else, the advice is the same.

It will be harder, for sure. Having a good advisor, active, in your research area can be extremely important, especially early on in your PhD, to make sure you are studying the right areas, and learning things that are relevant to current academic research. Without such an advisor, you will have to be a lot more proactive in learning things on your own, reading textbooks and papers from your research area, studying problems, and digesting material (e.g. blog posts) from some well-known researchers at other universities.

However, by no means is it impossible. And there are three points to make here:

  1. Having no professor in the subject X would be even harder, but here you merely have no active professor. That leaves you in a much stronger position. I would advise against being officially advised by a professor who is not active, but you can still have a close mentorship with them, and they can be your "real" advisor if they are interested and available. Just because they are not publishing anymore doesn't mean you can't learn a lot from them.

  2. Your passion and interest in a subject will take you far further than any mentorship. If you choose a subject that you are not interested in, then why do a PhD at all? To spend 5-10 years on something, you must truly love it, and truly want to become an expert. If you do want those things, you can succeed, although it is a lot of hard work. (Your interests may evolve later -- I find that almost all students' interests do, e.g. evolving more towards what their advisors do -- but if you are right now interested mainly in X, then you should do X.)

  3. By the end of your PhD, whether you have an expert in your field or not, you become an expert in your specific sub-problem who knows much more than your advisor. So, although choosing an area where you don't have an active advisor makes you "on your own" earlier, you would be on your own eventually anyway -- that is the whole goal of a PhD.

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