Physics influencing Mathematics

[PhDorBust]

If one wishes to look at where mathematics has influenced physics, there are an abundance of examples.

What are some examples of physics influencing mathematics? The development of calculus would be one certainly, but what about in the last century?

 

[homeomorphic]

Lots of examples.

A recent one would be the use of gauge theory in topology. Sieberg-Witten theory provides us a means of producing many different smooth structures on certain 4-manifolds (for example, by certain knot surgeries on the elliptic fibration E^2, which we have been discussing in our topology seminar here--this changes the Sieberg-Witten invariants without changing the topology, so it yields examples of topologically, but not smoothly equivalent manifolds). Many topologists might take it as a black-box, but it's based on such things are spinors, Dirac operators, gauge theory. Sieberg-Witten theory also appears to have been a big influence on such things as Heegard-Floer homology, which is a hot topic in topology, now. And of course, before Sieber-Witten, there was Donaldson theory, which might be thought of as a more primitive version of it. This was inspired by Yang-Mills gauge theory.

A related thing, which is what I study, is topological quantum field theory. It's very bare-bones quantum mechanics, I would say. It has some significance in pure math, but I think the interest in it comes more from physical considerations.

Another related thing is the Jones polynomial, which is a knot invariant. Jones was inspired here, originally, by studying Von Neumann algebras, but there were various connections to physics that were revealed later. The first math paper I ever read showed how the Jones polynomial could be thought of as some kind of partition function of a knot diagram. Also, Witten gave it an interpretation in terms of Chern-Simons theory and path-integrals.

Another closely related example would be the theory of quantum groups, which began as an effort (by physicists) to produce solutions of the Yang-Baxter equation from physics.

These are all pretty recent developments (last 30-40 years).

Another example would be the use of mirror-symmetry from string theory to solve certain enumeration problems in algebraic geometry.

The general theory of relativity influenced the development of modern differential geometry.