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I am a student of Mathematics who have to choose its area of specialization. I am trying to obtain as more information as possible, by asking a lot of questions to more experienced people, trying to have enough idea to make the right choice (or something really near). I should be very glad to anyone who should choose to spend a part of its time to help a poor student lost in its sea of possibilities :-)

My first (vague) question simply is:

  • why have you chosen to specialize exactly in the area X, instead of something different? What do you think about the remaining area? Let's assume that a student loves areas A,B,C in the same way...how could he choose the most appropriate? Do you have some general advice?

My second question is instead much more specific:

  • Let's assume that I love Category Theory and Homological Algebra, but I hate polynomials. Could be a wise idea to proceed towards Algebraic Geometry? I see that the by an elementary point of view, it seems to be strongly related to the solution of systems of polynomials (a problem in which I am not so interested), but on the other hand it proceed towards a good level of abstraction and use of CT. Is this level again related to polynomials, or are they "abandoned"?

Edit My question has been put on "on hold" because considered too subjective. I disagree, and suspect that it has been misunderstood . More precisely, my question could be reformulated as follows:

I know that polynomials play a key role for motivating the basic development of Algebraic Geometry. Are they still present/fundamental in the development/understanding of modern algebraic geometry, or are they maybe "obsolete" and "abandoned" in favor of more advanced ideas from Category Theory/some related area?

Thank you in advance for any help!!! Cheers!

Ps: I am not a native English speaker, consequently my apologies for some probably grammar-related errors.

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closed as primarily opinion-based by Grigory M, Servaes, Michael Galuza, vadim123, Kevin Carlson Aug 23 '15 at 20:04

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ I don't think you can avoid polynomials in general but I'm open to correction. You might find that you enjoy certain things about them if they help you prove something you are interested in? $\endgroup$ – snulty Aug 23 '15 at 9:48
  • $\begingroup$ I tell you what not in just in mathematics, if you like something you should know why you like it. If you hate it, why it is putting you out... and this reason gives some valuable information to you about yourself. $\endgroup$ – Narasimham Aug 23 '15 at 15:19
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    $\begingroup$ One often hears young math students saying they dislike X while contemplating future studies. At best, this shows immaturity and lack of professionalism (I have never heard a physician who exclusively specializes in one part of the body say that they hate another part). Taken as a long-term state: failure in mathematics is virtually guaranteed if you cultivate an active dislike of a broad and/or basic part of mathematics. Since you asked: doing algebraic geometry with a hatred of polynomials is indeed ridiculous...assuming you are using "hate" in anything like the standard way. $\endgroup$ – Pete L. Clark Aug 23 '15 at 18:41
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    $\begingroup$ "but surely I don't like and I prefer to avoid them as more as possible" What I'm trying to say is that to a mathematician, that sounds like a would-be writer averring that he will try to avoid using the letter "g" as much as possible. If you agree that your perspective lacks professional maturity, I would recommend that you talk to people who are more professionally mature, i.e., the faculty in your department. $\endgroup$ – Pete L. Clark Aug 23 '15 at 19:27
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    $\begingroup$ Finally, although there are branches of mathematics in which an eccentric hatred of polynomials could be relatively harmless, the OP has mentioned an interest in algebraic geometry, i.e., the branch of mathematics in which polynomials are the most basic object. The OP hopes that after enough study and abstraction algebraic geometry will eventually not be concerned with these distasteful objects, and the main point of my comment was to offer a professional perspective: no way. $\endgroup$ – Pete L. Clark Aug 23 '15 at 22:51