TITLE: Is learning (theoretical) physics useful/important for a mathematician?
QUESTION [77 upvotes]: I'm starting to read The Princeton Companion to Mathematics, at the beginning it says:

A proper appreciation of
  pure mathematics requires some knowledge of applied
  mathematics and theoretical physics.

Some of my professors have told me that modern Mathematics require some knowledge about Quantum Mechanics and theoretical Physics.
I attended the second and third day of the José Adem Memorial Lecture Series by Matilde Marcolli at the beginning of this year, they were about Number theory, Quantum statistical Mechanics and Quantum Field theory. I did not understand a single word but somehow the few things I understood had a big impression on me.
Please give me some examples of pure Mathematics that require/use Physics.

REPLY [4 votes]: There's a big difference between mathematical physics and theoretical physics.  The former is essentially a branch of mathematics.  A physicist would motivate, for example, generalized functions, by pointing out that ever real measurement involves an integral (average) of an "observable field" with respect to a "test function." While you can certainly master generalized functions without ever speaking of measurements, I can't help thinking you will have a limited perspective the subject.  
When I visited Cambridge, the pure mathematicians would send offprint request to the applied mathematicians by post.  But then, to the former, DAMPT was the department of awful mathematicians trying to be physicists.  Vive la différence.