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I've seen many courses labelled as 'mathematical physics', but I'm interested in knowing about the opposite: 'physical mathematics'.

I've noticed that some areas of mathematics which I found extremely abstract and not interesting were directly inspired by or popularized by physics: functional analysis, differential geometry, symplectic geometry and contact structures, partial and ordinary differential equations.

I've noticed, however, that students often dislike interrupting a mathematical course to introduce some motivating physics. So my question is,

What topics could be included into a 3-credit hour, one semester graduate course dedicated to describing physical systems and the math which results from them? Specifically, is there a set of topics that would form a somewhat cohesive whole while allowing enough time to get to interesting math?

One specific example I've thought of is doing the first few chapters of Dirac's Principles of Quantum Mechanics to teach functional analysis from a physics viewpoint and combining this with the first three chapters of Wald's General Relativity to teach differential geometry from a physics viewpoint. However, these topics are very different, and their is probably enough material in those combined chapters to teach several courses. What topics would be interconnected but not contain too much material?

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    $\begingroup$ What is the goal? To teach them some mathematics used in physics or to teach them some physics that uses advanced math? Would you like the audience to be more math grads or physics grads? $\endgroup$ – Chris C Jul 23 '14 at 14:07
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    $\begingroup$ If teaching physics to a mathematician, consider Michael Spivak's Physics for Mathematicians: amazon.com/Physics-Mathematicians-Mechanics-Michael-Spivak/dp/… $\endgroup$ – Chris C Jul 23 '14 at 14:08
  • $\begingroup$ @ChrisC Teaching advanced math to math majors through physics. $\endgroup$ – Brian Rushton Jul 23 '14 at 22:17
  • $\begingroup$ I really like the idea. Particularly when learning calculus and multi-variate calculus, all of the best applications were straight from physics. $\endgroup$ – Jeff-Inventor ChromeOS Aug 6 '14 at 18:47
  • $\begingroup$ incidentally, my thesis work on supermathematics is more or less all in the arena you describe. From a math perspective, I'm not sure I know a really convincing reason to study $\mathbb{Z}_2$-graded objects. But, from physics, it's as easy as knowing there are bosons and fermions and the mathematics of both fit best into a mathematics which alllows both commuting and anticommuting coordinates. For me, this means an infinite dimensional point-set built from Grassmann generators with finite norm, whereas most folks use some sort of non-commutative sheafy thing... $\endgroup$ – James S. Cook Mar 22 '15 at 19:49
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The Virasoro Algebra arises in the study of the quantum mechanics of strings. This is one of the fundamental examples which leads to the study of vertex operator algebras. When I saw the derivation in a string theory course it wasn't too lengthy.

Another topic would be exotic four dimensional manifolds. Donaldson Invariants arose from the study of instantons. The physics of instantons was used to discover the existence of four dimensional manifolds. I gather that the Seiberg-Witten theory has supplanted some of the Donaldson theory and it would also appear to fit the bill of physics being used to do mathematics. I believe that is the theme of Witten's work which impacts mathematics; some insight in quantum field theory has been used to do math.

At a lower level, since geometry is one of the oldest forms of physics, you can argue that physics is used whenever we resolve a proof by geometric intuition. For example, the proof of Rolle's Theorem; "what goes up must come down, so somewhere it had to stop going up and come down". That would be an example of using physics to prove mathematics. I say "prove" in an very relaxed sense of the word!

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  • $\begingroup$ admittedly, my answer is hard to use for undergraduates. My best student just barely made it to classical gauge theory and it takes a full undergrad in both math and physics to really appreciate the application of QFT to math... maybe it takes a PhD in physics... $\endgroup$ – James S. Cook Jul 24 '14 at 14:12
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I think you might appreciate an example of what has been tried and can be done.

I recently took a one-semester seminar-type course in mathematical general relativity, servicing both mathematics and physics students, and I would say I learned quite a lot. We met 80 minutes per week with the students (undergrads) preparing and giving the lectures with guidance from the organizer (faculty). The first half of the course was devoted to developing the language of Riemannian manifolds, mostly rigorously but tiptoeing past the more abstract definitions. The second half focused on general relativity; we were able to formulate the Einstein field equations and discussed the Schwarzchild solution toward the end, and a quick description of Penrose diagrams and what black holes are. Given more time (say as a proper course meeting twice a week) we would probably have been able to talk about geometric analysis (developing the language of analysis along the way).

Courses like this can be quite successful. I think quantum mechanics is a great subject for mathematicians and physicists to learn from together; statistical mechanics is probably another, and I think fluid dynamics could be done (maybe mathematically from the dynamical systems perspective, or perhaps the analysis of Navier-Stokes).

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