Manchester Geometry Seminar 1999/2000

The talk will start with a gentle introduction to the topologist's version of the theory of toric varieties, which was until recently the provenance of algebraic geometry. One of the prime subtleties in the topological version is the absence of underlying complex structure, and the second half of the talk will address ways in which weaker structures may, nevertheless, be associated to toric manifolds. Since we must keep track of at least 2^(n+1) orientations for each manifold of dimension 2n, the speaker will concentrate on surveying the philosophy, rather than providing precise details!