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I am currently in my final semester of masters. I have some background in Algebraic Number Theory. I have one PhD offer from a US university.

When I started getting into algebraic number theory, I did not have much trouble as most of the pre-requisites were from Galois Theory and Commutative Algebra. But through my study in the last two-three semesters, I have realized that Number Theory is a subject which requires knowledge of many different areas like Complex Analysis, Algebraic Geometry, Topology(subjects which I have not learned properly or not learned at all). This has put me under a quite a lot of stress: Am I good enough to study Number Theory or should I shift to something else, something more self-sufficient?

Is it possible (and advisable) to change my research interest to something else? What should I do?

Edit- To add to my troubles, I don't know anything about Modular Forms or Automorphic Forms or Galois Representations.

Before voting to close this question down, I would like you to know this: I originally asked this question on Math SE. But people told me that it was more of an Academia SE question.

  • Is the offer for a PhD in number theory? If it is then it is probably wirth going for it. Yes, you will need to learn a bunch of topics that you may not feel that comfortable with right now, but you will have an advisor who can guide you on the order in which to learn these in a way that can at the same time take advantage of the knowledge you do have in other fields. I did something similar. I focused on finite group theory during my master's then got a PhD with a topic in the representation theory of algebraic groups and quantum groups. – Tobias Kildetoft Mar 28 '17 at 8:11
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You haven't mentioned any alternatives to Number Theory. Hmm. Maybe it would be reassuring for you to have a small list in your hip pocket of possible areas.

If you don't have to decide right away, then I would suggest that you hold off on deciding, and meantime start remedying these gaps as soon as possible. Without panic, but without any shilly-shallying. Obviously, don't work on them all at once, but do start somewhere, and make a tentative map through this subject matter -- in what order, what books will you use, what topics could be skipped or skimmed over, which ones will you do in a class, which as self-study, and which in an independent study.

Even if you end up working on something other than Number Theory, this study will be helpful for you in the long run, either directly or indirectly.

If it feels daunting, think one step at a time, or Rome wasn't built in a day, or some similar phrase that helps you.

In the U.S., one is generally not expected to hit the ground already doing research, right from the first semester. It's okay to take some time and get the solid foundation one needs.

Do you have an advisor, potential advisor, temporary advisor, or mentor of some sort? If not, look around for one, and if necessary ask the department to assign you one temporarily.

  • many areas of math use at least one of topology, complex analysis, or algebriac geometry. – Rüdiger Mar 27 '17 at 23:32
  • @Rüdiger - Are you agreeing with me ("this study will be helpful for you in the long run, either directly or indirectly"), do we differ in some aspect of this, or are we having a misunderstanding, perhaps? – aparente001 Mar 28 '17 at 0:42
  • I agree, +1. I am just emphasizing this point once more. It is very important that the OP learns the subjects in question.... any mathematician worth the name knows the basics to some degree. – Rüdiger Mar 28 '17 at 9:59
  • @Rüdiger - Thanks for clarifying, I understand now. – aparente001 Mar 29 '17 at 1:15
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If you are an intellectually-oriented person, I would advise you to first and foremost follow your own genuine interests.

Another relevant concern is career planning: what do you want to work on the rest of your life? Your education should enable your desired career.

There is also the trade-off between money (wage), type of work, and general lifestyle.

In fact, there are so many concerns to consider that you are likely to change plans as you go. Therefore, aparante's suggestion of taking it one step at a time is an excellent one.

If you have a strong and genuine interest in the program of study that have been offered to you, I'd suggest you try it out. Don't underestimate yourself or overestimate others. Just have a go and see what comes out of it. That way you'll have no regrets later.

However, if this offer is more of a coincidence, or something someone else has pushed on you, examine your own interest in it carefully and let that be your guide.

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