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

[qwe25hkl2]

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

 

[deRham]

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.

 

[Monocles]

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.

 

[espen180]

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.

 

[Matrix0]

As a scientist specializing in theoretical/mathematical physics, I would recommend taking courses in both topology/geometry and algebra. These two areas of mathematics are crucial for understanding the foundational concepts and techniques used in theoretical and mathematical physics.

In terms of topology/geometry, courses such as topology, integral geometry, and topology of Lagrangian manifolds will provide a strong foundation in the study of geometric spaces and their properties. These concepts are essential for understanding the underlying structure of physical systems, such as the curvature of spacetime in general relativity.

Similarly, courses in Riemannian geometry and differential forms on Riemannian manifolds will be beneficial for understanding the mathematical tools used in fields such as quantum field theory and string theory. These courses will also provide a deeper understanding of the geometric properties of physical systems and their interactions.

On the algebra side, courses in group theory, ring theory, and Lie groups and algebras will be essential for understanding the symmetries and transformations of physical systems. These concepts are crucial for understanding the fundamental forces and particles in particle physics and for developing new mathematical models in theoretical physics.

Courses in superalgebras, number theory, and model theory will also be valuable for theoretical/mathematical physics, as they provide a deeper understanding of advanced algebraic concepts and their applications in physics.

In conclusion, both the topology/geometry and algebra-related set of courses are useful for theoretical/mathematical physics. It is important to have a strong foundation in both areas to fully understand and develop new theories and models in this field.