In India, generally during the graduate years, we follow a course work pattern, unlike many places in the U.S where students are exposed to research during their undergraduate years itself.

As, a graduate student, we have basic courses like, Algebra, Analysis, Rings and Modules, Measure theory, Topology, functional analysis etc.. Generally topology is one subject, which i don't find that much of interest. But in some universities, students are forced to take Topology, and Algebraic topology, during their graduate years, and i have seen many students facing trouble, as they have to study a subject which is not of their interests. My question, would be, for a student whose research are in Analytic and Algebraic Number theory, does he needs to know Algebraic Topology?

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    $\begingroup$ To study number theory, one needs to know as much mathematics as possible. These days there's a lot of cohomology around, and one first meets cohomology in algebraic topology. $\endgroup$ – Robin Chapman Sep 13 '10 at 14:43
  • $\begingroup$ @Robin Chapman: There are no easy measures right! OK, I dont see algebraic topology coming into any effect atleast in analytic number theory! $\endgroup$ – anonymous Sep 13 '10 at 14:46
  • $\begingroup$ @Chandru1, what Robin has in mind, probably, is the fact that "cohomological ideas", which one usually first encounters in the context of algebraic topology (and this is a good thing, really, because there they are most palpable), are surely relevant to all forms of number theory nowadays. $\endgroup$ – Mariano Suárez-Álvarez Sep 13 '10 at 14:53
  • $\begingroup$ Chandru, I didn't know you were a professor...congratulations. If you stick to the most "hard-line" analytic number theory, you won't see much cohomology, but if to you analytic number theory embraces modular forms, or asymptotically counting solutions of Diophantine equations, you won't be able to avoid it. $\endgroup$ – Robin Chapman Sep 13 '10 at 15:07
  • $\begingroup$ @Robin: Yes, equations, of the that form are related to somewhat study of elliptic curves where Galois cohomology plays an important part! is this what you want to say $\endgroup$ – anonymous Sep 13 '10 at 15:12

Well, I'm not an expert in Number Theory and I don't know if your interests may include in the future things like Weil conjectures and $\zeta$-functions for algebraic varieties, or on the contrary, you'll try to avoid any contact with Algebraic Geometry and Homological Algebra.

But, just in case, your interests lead you towards the first issues, then you'll have a nasty encounter with something called $\ell$-adic and étale cohomologies. I wouldn't like to be in your shoes at that moment, without having seen before any other simpler cohomology (as the singular cohomology you're going to learn in Algebraic Topology) and the classical Lefchstez fixed-point theorem.

More generally, except you're not going to use Algebraic Geometry at all in your research, nor Homological Algebra, or your use of the first is limited to the most classical aspects, you'll have to be familiar with sheaf cohomology and derived functors such as $\mathrm{Ext}$ and $\mathrm{Tor}$. It isn't impossible to learn sheaf cohomology without knowing a word of singular cohomology, and the same applies for derived functors, but you'll clearly have a tremendous gap at that point.


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