Abstract: One might expect the space of holomorphic curves on an infinite-dimensional complex manifold to be too huge to understand well. But in certain cases -- when the manifold is a quotient of an affine Kac-Moody group by a parabolic subgroup -- Atiyah observed that the based holomorphic curves are parametrized by a finite-dimensional space. We will explain how to compactify this space and study its topology using a construction similar to the "stable maps" of Kontsevich.