Alexandrov Geometry of Leaf Spaces and Applications

We develop a number of tools to analyze the geometry and topology of leaf spaces - quotients of singular Riemannian foliations with closed leaves. This expands upon and gives a purely geometric footing to similar tools used to study orbit spaces of isometric group actions. When applied to a given leaf space, these tools not only help describe the geometry/topology of the quotient, but can also reveal information about the leaves of the singular Riemannian foliation and the manifolds which admit such foliations. The majority of the work is done for leaf spaces with positive curvature (in the comparison sense), as the original motivation was to systematically study singular Riemannian foliations as was done for positively curved manifolds with symmetry.