I'm a physics and math major, going into my 3rd year. Suppose I want to do research in theoretical aspects of condensed matter. What would be the mathematics I should be learning as an undergraduate? Here is a rundown of courses I'm considering taking next year:
- Abstract Algebra: it seems a little preposterous for a math major to not have taken a course in abstract algebra, and things like group theory seem to be very applicable in theoretical physics from what I have read.
- Differential Geometry: from what I've read, differential geometry is not used in condensed matter, except for a few fancy things like cell membranes.
- Topology: it looks like topology is used extensively, although it isn't clear to me if taking a upper-undergrad course in point-set topology will be of any use. It looks like it isn't until one learns algebraic topology that some of the concepts (homotopy, homology etc.) would become of any use, and even then I'm not sure if physicists would learn those at the same level of rigour as a mathematician would. If I wanted to focus on topological effects specifically, would it be advisable to take a graduate sequence in algebraic topology, or take an undergraduate course in point-set topology, or neither?
- Linear Algebra: I have the option to take a more advanced proofs-based linear algebra course which mostly covers topics I haven't encountered in the first course I took (canonical forms, adjoints, operators, spectral theorem). I liked linear algebra and have seen some people comment to take as many linear algebra courses as possible for condensed matter / theoretical physics.
- Transform Methods: there is also one course that covers a wide range of transform techniques in depth (Fast and Discrete Fourier, Laplace, Radon) and their applications, and I'm not sure how useful these would be.
- There is also a course on chaotic and non-linear behaviour offered and I would have thought it would be a great course to take for condensed matter as it also touches on collective behaviour.
- Differential forms: I know nothing about this one.