A construction of semi-infinite de Rham cohomology
The purpose of this thesis is to describe a construction of semi-infinite de Rham cohomology for infinite dimensional manifolds equipped with the extra structure of a polarisation. We describe the construction for finite dimensions and show how it extends to other cases; in particular the semi-infinite. We then define variations for Hilbert manifolds which allow us to calculate the semi-infinite cohomology of the projective space and the Grassmannians of a polarised Hilbert space. Finally, we consider some of the implications of these results for index theory, in particular for the Witten genus.