I'm an undergrad interested in theoretical physics. Don't know enough to decide which specific subfield I may go in, but particle theory (strings, QFT, GR) and mathematically heavy condensed matter seem appealing.
I have taken quite a few pure math courses in my undergrad career so far (algebra, analysis, topology etc), but now that I've exhausted most of the undergraduate curriculum, its time to start some advanced topics such as Algebraic Topology and Lie Algebras etc. My concern however is that these courses are offered by the mathematics department and thus the exposition and focus is aimed towards training mathematicians rather than someone who may find them useful in another field. So as expected in such a class, the main goal is to understand and prove the main theorems of these fields. However, as a physicist I'm not sure to what extent I can benefit from such an exposition. It would be highly time consuming to take such an approach to mathematics for someone to which mathematics is not the ends itself, but rather the means. On the other hand, there is the option of using "math for physics" type books such as Nakahara or Frankel to learn these topics, which would be a lot quicker and would have a physicist as a reader in mind, but no doubt, depth of understanding would be lost with such an approach. So my question is:
Is it worth it for an aspiring theoretical physicist to learn advanced mathematical topics from graduate math classes (and as a result sacrificing the time spent physics), or would it be a better use of his time to focus on the physics and pick up the math from a text or a class of the "for physicists" type? How would the answer change if applied to a mathematical as opposed to theoretical physicist? Note that over here I'm talking about topics such as Algebraic Topology, Differential Geometry, Representation Theory etc.