Starting from 80-ies the ideas either coming from physics, or by physicists themselves (e.g. Witten) are shaping many directions in mathematics. It is tempting to paraphrase E. Wigner, saying about "The Unreasonable Effectiveness of Physics in Mathematics".
What can be the reasons for it ?
Do physicists have some tools/ideas/techniques which allow them to make insights, which are not seen for mathematicians? Or it is just because Witten&K are very ... very smart ?
If, yes, what are these tools/ideas ? How to learn/absorb/(put into math. framework) them?
What can be the further applications of these ideas ?
Being a mathematician, but working in physicists surrounding for many years, I have thought on this questions for quite a while. The recent MO question http://mathoverflow.net/questions/116251/mathematician-trying-to-learn-string-theory/116402 prompts me to ask it here.
I would think, that yes, there are such "ideas". But from some outstanding mathematicians I've heard an opposite opinion.
My vague feeling is that quantum field theory and string theory it is something like an analysis/differential geometry on infinite-dimensional manifolds. But these manifolds are not abstract, say Banach modeled manifolds, which theory is not so rich, but kind of maps from one finite-dim. manifold to another, which has certain specific structures which are not fully revealed by mathematicians. E.g. vertex operator algebras, arise from maps of circle to manifold, if we map not circle but something higher dimensional there should be something more complicated. Another issue is about Feynman integral, which allows physicist to use integration techniques in geometric problems, it is not well-defined mathematically, but it might be it cannot be defined in very general form of infinite-dimensional integrals, but again physicist have an intuition where it can be defined, where cannot, and proper mathematical theory should first clarify the setup where it exists, rather than trying to build general theory which might not exist. These words are probably very vague, so might be answers help to me clarify.
I think everybody knows the influence of physics happened from 80-ies, but for completeness let me mention just a few.
Donaldson used instanton moduli spaces in his study of 4-folds.
Faddeev, Drinfeld et. al. created quantum groups
Representation theory of infinite-dimensional algebras have been large influenced by conformal field theory developments.
Witten's contributions are numerous his Fields Medal says more than I can say.
Mirror Symmetry, quantum cohomology etc...
The works of Fields Medalist Kontsevich and Okounkov are largely influenced by physics.
So on an so forth...