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I am a physics student working on Yang-Mills theory that is basically based on Manifolds, Geometry and some other regime of mathematics. I have heard Algebraic Geometry has various usage in physics. My question is concerning about which one, Riemannian Geometry, Algebraic Topology, or Algebraic Geometry has more practical usage in mathematical physics, especially in geometric approaches toward modern fundamental theories.

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  • $\begingroup$ Between those I would say Riemannian Geometry. To get to a point where you absolutely need to know algebraic topology/geometry you would have to be studying really particular things. I would say that the theory of fiber bundles has more practical use than those, since gauge theory is what QFT is based in (as the standard model). $\endgroup$ – Grassy LittleRoot Jan 25 '17 at 9:07
  • $\begingroup$ @CapimMatinho honestly I have to pick up courses until tomorrow and I am unsure about Riemannian Geometry and Algebraic Topology. Which one would you recommend? $\endgroup$ – mathvc_ Jan 25 '17 at 15:01
  • $\begingroup$ well, yang-mills is gauge theory... I don't know if it utilizes either very much. If you're already familiar with basic differential geometry I would do the algebraic topology course, because imo a course in riemannian geometry would be good for getting familiar with basic dg, even if you're not really going to use the "riemannian" part. I'm no expert in any of areas (still undergrad), so I would wait for someone qualified to answer... but that's what I would do. $\endgroup$ – Grassy LittleRoot Jan 25 '17 at 15:25
  • $\begingroup$ @CapimMatinho I've already passed differential manifolds. I guess I would take algebraic topology course. $\endgroup$ – mathvc_ Jan 25 '17 at 15:32

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