It introduced a new geometric structure called a generalized complex structure, which on the one hand unifies symplectic and complex geometry, something which mirror symmetry does at a more sophisticated but less understood level, and on the other hand provides a natural framework for other geometric structures of interest in string theory.

Both. The starting point is the study of an invariant functional in six dimensions which, together with a simpler, earlier one, has been seen as originating topological M theory. Yet, it also opened the window on the existence of generalized structures in all dimensions, a subject further developed with my students. In doing so it introduced a new basic scenario for doing differential geometry which incorporates the physicists’ B-field.

String theory provides a wonderful resource of ideas which are useful in mathematics, but they are rarely formulated in terms which the mathematician can understand, or provide rigorous proofs for. Sometimes the impact goes the other way, and this seems to be one of those occasions. It is a simple mathematical idea which explains and sets in a common context some of the structures and rules introduced by string theorists.

I began by being interested in differential geometric structures defined by closed forms, and in particular whether the moduli space was determined by the cohomology class of the form. In particular, motivated by what the physicists knew, I wanted to understand in my own terms the moduli space of Calabi-Yau threefolds. This led me to the invariant functional approach and the study of open orbits of Lie groups. Having written papers on symplectic, Calabi-Yau, and G2 geometry in this form, I noticed during a lecture on an apparently different topic an open orbit with similar properties. In attempting to describe a type of geometry that fitted in with this I found the B-field and all the attendant structures appearing very naturally.