Topology of moduli spaces

There are many ways of forming spaces of mathematical objects, or of subobjects of some fixed one, and it is usually of considerable interest to examine the topology of such spaces. Examples are moduli spaces of smooth manifolds, giving a new perspective on the study of diffeomorphism groups, classical examples such as configuration spaces and their variants, or spaces of Riemannian metrics subject to curvature conditions. Increasingly sophisticated techniques from homotopy theory are being used to examine such spaces. This project will deploy such methods to investigate questions such as: can distinct mathematical objects have equivalent moduli spaces?