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.