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Just curious: I can't think of a single example that a physicist did not had his mouth open in amazement when he learnt that all (OK, lets say the foundations) the math he needs for his brand-new theory had already been paved out long before by mathematicians, possibly just for the fun of it.
Stupid me, of course I can think of an example, I'm always doing it...Knot theory really began with the physical theory of knots in "ether".

Probably should be tagged "small list" but we don't have that one :-)

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    $\begingroup$ Could you clearly state your question please? $\endgroup$
    – Benoît Kloeckner
    Commented May 11, 2015 at 18:36
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    $\begingroup$ How about calculus by Newton? Or Dirac's delta function? $\endgroup$
    – KConrad
    Commented May 11, 2015 at 18:39
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    $\begingroup$ Unfortunately I am not competent enough to make it an answer but I strongly believe that mathematicians still have not found truly adequate formalism to deal with things like chiral algebras, vertex operators, operator product expansions... VOAs have made very interesting piece of mathematics but in my opinion it is still quite obscure for general mathematical community $\endgroup$
    – მამუკა ჯიბლაძე
    Commented May 11, 2015 at 18:50
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    $\begingroup$ There are plenty of examples when new math was created by physicists or for the needs of physics. A VERY large part of Mathematics was created to answer the concerns of physicists. $\endgroup$
    – Alexandre Eremenko
    Commented May 11, 2015 at 18:56
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    $\begingroup$ Physicist to string theorist: You've had this research program going on for 35 years, and it's not yet looking like a healthy physical theory. Why should anyone keep giving you funding? Standard answer from a string theorist: String theory is a piece of 21st-century mathematics that's fallen out of the sky into the 20th century, and it's going to require 22nd-century mathematics to solve it. Even if it's never going to be a correct Theory of Everything, keep funding us, because it's wonderful mathematics. $\endgroup$
    – user21349
    Commented May 12, 2015 at 2:17

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I would suggest chaos theory as originating from examples of chaotic systems in physics. As described here, there are plenty of mathematicians among the pioneers (Poincaré, Kolmogorov,Lorenz,Feigenbaum...), but the motivation came from physics (planetary motion, turbulence,weather,population dynamics...).

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    $\begingroup$ I would rather argue that there is no such thing as "chaos theory" except perhaps in the newspaper and Hollywood movies. $\endgroup$
    – Christian Remling
    Commented May 11, 2015 at 19:07

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