Choosing math courses for theoretical/mathematical physics (grad school)

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Discussion Overview

The discussion revolves around the selection of mathematical courses that are beneficial for theoretical and mathematical physics, particularly in the context of preparing for graduate school. Participants explore various topics within topology, geometry, algebra, and their relevance to the field.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • Some participants suggest that courses in topology and geometry, such as topology of Lagrangian manifolds and Riemannian geometry, are important for mathematical physics.
  • Others mention that superalgebras and various algebraic structures are relevant, particularly in contexts like supersymmetry.
  • One participant notes that model theory and number theory have limited applications in mathematical physics, although they acknowledge some specific uses for number theory.
  • Another participant emphasizes the necessity of understanding the entire theory of smooth manifolds, not just Lagrangian manifolds.
  • There is a suggestion to include measure theory and mathematical analysis as additional important areas of study.

Areas of Agreement / Disagreement

Participants generally agree that most of the listed mathematical topics are relevant to mathematical physics, with some exceptions noted for model theory and number theory. However, there is no consensus on the extent of usefulness for each course or topic.

Contextual Notes

Some limitations in the discussion include the lack of clarity on specific applications of certain topics, such as integral geometry, and the varying degrees of familiarity participants have with the subjects mentioned.

Who May Find This Useful

Students and professionals interested in pursuing theoretical or mathematical physics, particularly those preparing for graduate studies, may find this discussion relevant.

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Which of these mathematical courses are useful for theoretical/mathematical physics?

Topology/Geometry-related set of courses
Topology
Integral geometry
Topology of Lagrangian manifolds
Riemannian geometry
Differential forms on Riemannian manifolds

Algebra-related set of courses
Group theory
Ring theory
Lie groups and algebras
Groups and geometries
Superalgebras
Number theory
Model theory
 
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I'm not quite a mathematical physics guy, but I know just a thing or two about what some of it entails. Things like superalgebras and all of the topology and geometry stuff seem to come up in a lot of mathematical physics for sure.
 
Basically all of those topics crop up in mathematical physics somewhere, with the possible exception of model theory (I've never heard of it being used in physics but that doesn't mean that it hasn't). Number theory also has very few uses in mathematical physics, but they do exist (one example: using quantum statistical mechanics to study class field theory). Superalgebras generally crop up in supersymmetry, but I don't know if they have much use outside of that. I don't know anything about integral geometry so I can't comment on that. Everything else you mentioned, though, is widely used in mathematical physics.
 
I think everything you listed except for model theory is used on a regular basis. I also think you cannot limit yourself to Lagrangian manifolds. You need the entire theory of smooth manifolds.

I would also throw in some measure theory and mathematical analysis.
 

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