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.



The influence of string theory on mathematics for philosophers.
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](http://www.mat.ufmg.br/~israel/jojoEE.pdf)


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).


