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Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How true this is, I am not sure.

So I suppose I want to ask, what sort of work do modern day algebraists do?

  • What are currently some of the more active areas of modern algebra?

  • What types of problems do algebraists deal with?

  • I'm kicking around the idea of pursuing graduate study one day, possibly in some sort of algebraic field, i.e., ring theory or something. What sort of research and problems are open to your average graduate student in algebra (of any sort, not just ring theory)?

This is partially inspired by the question What do modern-day analysts actually do?

Thank you for your responses.

  • 6
    It is not clear to me that there is a useful distinction between algebraists and algebraic geometers, algebraic number theorists, algebraic topologists, ...2012-08-11
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    @QiaochuYuan I think we are getting to a point where the main distinction is motivation.2012-08-11
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    @QiaochuYuan You are missing Hwang's point, she was probably hinting towards "pure" algebraists (Hopf algebras, Quantum groups, ring theory, group theory, generally non-commutative "stuff")2012-08-11
  • 1
    @Chindea: it is also not clear to me that quantum groups count as "pure" algebra.2012-08-11
  • 1
    @Yuan Well, there are some people "out there" having something of Midas' "golden touch". Still, it's left to the OP to agree with the lack of a "useful" distinction between completely different areas of mathematics, as you suggest2012-08-11
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    I don't see a clear distinction either, but the people that I know who seem closest to "pure algebraists" are actually model theorists. But there are also those who deal with non-commutative harmonic analysis, which could be interpreted as "purely algebraic", if you follow Chindea Filip's comment...2012-08-12
  • 0
    @TiffanyHwang: As someone who does research in algebra, I often feel the same way...2012-08-13

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