The influence of string theory on mathematics for philosophers. I've agreed, perhaps unwisely, to give a talk to Philosophers about string theory.
I'd like to give the philosophers an overview of the status and influence of string theory in physics, which I feel competent to do, but I would also like to say something about the influence it has had in mathematics where I  am on less
familiar ground. I've read the Jaffe-Quinn manifesto and the responses in 
http://arxiv.org/abs/math.HO/9404229.  What I would like from MO are pointers to more recent discussions of this issue in the mathematical community so that I can get a sense of where things stand 16 years later.
 A: Dear Jeff, string theory has had a colossal influence on the renewal of enumerative geometry, a two century old branch of algebraic geometry inextricably linked to intersection theory.
Here is a telling anecdote.
Ellingsrud and Strømme, two renowned specialists in Hilbert Scheme theory, had calculated the number of rational cubic curves on a general quintic threefold by arguments based on their paper 
On the Chow ring of a geometric quotient, Annals of Math. 130 (1987) 159–187
Their result differed from that predicted by string theory. Of course everybody thought the mathematicians were right, but actually there had been a programming error in their calculations and the correct result was that of the physicists (which Ellingsrud and Strømme confirmed after fixing their bug). 
This was the beginning of a long list of results predicted by string theorists and subsequently proved by mathematicians, a celebrated example being Kontsevich's formula for the number $N_d$ of degree $d$ rational curves in $\mathbb P^2$ passing through $3d-1$ points in general position. 
You can read all about Kontsevich's formula in Kock and Vainsencher's free on-line 
book
And a pleasantly elementary general reference is Sheldon Katz's Enumerative Geometry and String Theory, published by the AMS in its Student Mathematical Library (vol. 32).
A: The Jaffe-Quinn manifesto really had little to do with string theory, but a lot to do with topological quantum field theory, especially 3d tqft.  I remember Frank Quinn talking about this at length during a hike at the 1991 Park City summer school.  He was lecturing there on topological qft, see
"Lectures on Axiomatic Topological Quantum Field Theory" in "Geometry and Quantum Field Theory, IAS/Park City Mathematics Series, Volume 1", edited by Daniel Freed and Karen Uhlenbeck.
The sort of thing that was worrying Quinn was:


*

*Witten's great paper on "Supersymmetry and Morse Theory", which was published in a mathematics journal, the Journal of Differential Geometry.

*Witten's Fields medal winning work on the Jones polynomial and Chern-Simons theory.


Quinn explained that at the beginning of his career he had been heavily influenced by the work of Thurston and Sullivan, but found that trying to emulate them had led him to lose track of what he precisely understood and what he didn't, requiring a painful period of getting back to a more rigorous way of working.  He was worried that losing the distinction between works like Witten's and truly rigorous work would lead others to the problematic situation he had found himself in as a young mathematician.  In the end, I think Atiyah's response won the day: he argued that mathematicians were fully capable of protecting their virtue while interacting with physicists. Shortly after this exchange, those topologists in the math community who were skeptical about the importance of what Witten was bringing to mathematics were conclusively won over by the Seiberg-Witten equations.
But the example set by Quinn of how to do TQFT in the end has largely won out.  There was an attempt to teach mathematicians the actual QFT behind Seiberg-Witten at the IAS in 96/97, but I don't think it was very successful.  These days both TQFT and the Seiberg-Witten equations remain very important ideas in topology, but they're pursued with conventional standards of rigor. Mathematicians have gotten used to taking physicist's QFT arguments and extracting and generalizing those parts that can be made rigorous and fit into the evolving mathematical tradition.
As others have mentioned, for the case of string theory, mirror symmetry is probably the best example of an idea coming out of it that has had a huge influence in mathematics.  Yau's recent popular book "The Shape of Inner Space" contains lots of other examples of the interaction of math and physics surrounding Calabi-Yau manifolds.   
A: One discovery in string theory that might be mentioned because it has had quite an impact on mathematics is mirror symmetry. While this discovery in 1989 predates the manifesto, it probably would fit into a mathematically oriented discussion because it was quite unexpected by mathematicians (and physicists), and its deeper mathematical ramifications became clear only later on.
