Rings of Conditions of Rank 1 Spherical Varieties

Abstract

In this thesis, we define and describe the rings of conditions of rank 1 spherical homogeneous spaces G/H. A procedure for computing the ring of conditions of a spherical homogeneous space in general is not known. For the special case of rank 1 spherical homogeneous spaces, we give a proof of the unpublished result of A. Khovanskii that the ring of conditions is isomorphic to the cohomology ring of a certain compactification of G/H. We illustrate this result through the fully worked example of affine n-space minus the origin.