Mathematics for a physics undergraduate

[woody4064]

Hello,
I have just finished my second year of a theoretical physics degree (MPhys) at the University of Nottingham in the UK. I now have three months with relatively few commitments, and would like to learn some additional maths that would be useful, but am unsure what topics I should study. Given that I do a lot of general relativity over the next two years, and I hope to do a PhD in particle theory, could anyone give me advice?

 

[Student100]

Differential geometry might be beneficial.

 

[WannabeNewton]

Hello,
Given that I do a lot of general relativity over the next two years, and I hope to do a PhD in particle theory, could anyone give me advice?

What do you already know?

 

[woody4064]

What do you already know?

Nth order ODEs with constant coefficients, Fourier transforms, using Fourier and separation of variables to solve PDEs, very basic stuff on tensors (as in, I know what they are, I know tensor notation but the only one I've ever used greater than order 2 is the Levi-Civita tensor), generating functions, vector calculus, and enough calculus of variations to derive the classical Lagrangian. Fairly typical stuff for a second year theoretical student.

 

[WannabeNewton]

Presumably if you've seen Fourier series solutions to PDEs then you've also seen solutions to simple Sturm–Liouville systems using Legendre polynomials, spherical harmonics, and Bessel functions, particularly in the context of Frobenius' method in QM or Laplace's equation in EM? If not then you might want to pick that up from a book on mathematical methods in physics (e.g. Arfken). If you've gotten that down then make sure you also spend some time on contour integration and the residue theorem/Laurent series; you will also find this in Arfken.

I'm also assuming you know linear algebra inside and out at this point, which you probably do given that you're second year. Then as far as general relativity goes, the only thing you would really need to spend time on with regards to math is differential geometry. You can go ahead and try to learn it from a pure math book if you wish-you'll find tons of threads here and elsewhere suggesting books for that. Or you could just start studying GR now and pickup the differential geometry as you go along since all GR books will have a few chapters on the necessary aspects of it.

The only other thing I can think of that would be of great use to you is representation theory, if your interest is HEPT. You can again go the pure math route here but you would probably be much better served going through a book on rep theory that has a physics student in mind. Some time ago the user dextercioby recommend to me the following: https://www.amazon.com/dp/3211834435/?tag=pfamazon01-20 and I've found it to be absolutely brilliant so you should check it out.

Good luck!