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Are Commutative Algebra and Algebraic Geometry useful for physics?

  1. Aug 9, 2013 #1
    Are "Commutative Algebra" and "Algebraic Geometry" useful for physics?


    I'm considering taking Commutative Algebra, and perhaps even Algebraic Geometry (for which the previous is a prerequisite). In the first place I would take it for the enjoyment of mathematics and to give me an all-round mathematical education (even though I'm not into abstract algebra, leaning more toward analysis-type stuff, think differential geometry etc).

    But I was wondering, is there also a value in, say, commutative algebra from a physicist point of view? (I suppose if one goes deep enough in string theory, one can find it being used somewhere but I hope you understand I'm (in the first place, at least) looking for a less terribly-specific application.)

    Thank you.
  2. jcsd
  3. Aug 10, 2013 #2
    Hmmm, only place I'VE seen it kind of used has been quantum gravity papers, but those were dealing with algebraic geometry approach to the theories. From what i remember is you'll mainly be dealing with noetherian rings, and the "expanding" the idea to things like Krulls principle, Hilberts theorem, and things of that matter.

    You'd probably enjoy the course if you like math, but if you want some physical application I think you'd benefit more from taking a class that deals with C* algebras than commutative algebra/algebraic geometry. You should find C* algebra being presented in functional analysis courses.

    Good luck!
  4. Aug 10, 2013 #3
    I'm sure it depends on what you are interested in (physics-wise). A couple of my instructors last year do research in mathematical physics and are algebraic geometers. I know group theory is useful for at least some people studying physics.
  5. Aug 11, 2013 #4
    Yes it can be valuable... I'd assume most people working in theoretical physics would have at least basic knowledge of such topics as commutative algebra and algebraic geometry: they can pop up all over the place. The mathematical physics program at my university actually requires students to take at least some abstract algebra (although it isn't emphasized as much in physics it is incredibly useful to have around!)
  6. Aug 13, 2013 #5
    Sure, but it depends on what you are trying to do. For example, algebraic geometry is completely unavoidable in string theory. Calabi-Yau manifolds are a certain special type of complex variety. Mirror symmetry is of great interest to both string theorists and algebraic geometers. Even stacks can arise - this paper gives an excellent overview: http://arxiv.org/abs/hep-th/0608056

    It can sometimes come up away from string theory as well. For example, Grothendieck's motives have a connection to (massless, scalar) perturbative quantum field theory.

    A lot of algebraic geometry used in physics seems to be of more interest to mathematicians (I have never met a physicist who computed a Feynman amplitude from the period of a motive), but I could just be biased since I do math.
  7. Aug 13, 2013 #6


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    Allow me to be of dissenting opinion.

    If you want to take it for the pleasure of learning mathematics, that's wonderful.

    If you do experimental physics, it will be of zero use to you.

    If you do theoretical physics, I would say the odds are still very stacked against you ever using it.
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