One of the stranger developments of recent years is the influence of physics on algebraic geometry. A dramatic recent example is Gromov-Witten theory, which draws its inspiration from quantum field theory, but can be used to study the moduli space of complex algebraic curves.

A moduli space is a space that parameterizes all objects of a certain type. The classic example is the projective line, which classifies lines in the plane: each line in the plane corresponds to one and only one point in the plane. The moduli space of curves classifies complex-algebraic curves. The space itself is a geometric object, but its structure turns out to be very complicated, and recent progress has relied on these ideas from physics.

Ravi Vakil has posted a survey article on the subject, The moduli space of curves and Gromov-Witten theory, to Arxiv. He also has an older, more elementary article from the June/July 2003 Notices of the AMS, The moduli space of curves and its tautological ring.