Doctor of Philosophy
Smooth manifolds are central objects in mathematics. However, the category of smooth manifolds is not closed under many useful operations. Since the 1970's, mathematicians have been trying to generalize the concept of smooth manifolds. J. Souriau's notion of diffeological spaces is one of them. P. Iglesias-Zemmour and others developed this theory, and used it to simplify and unify several important concepts and constructions in mathematics and physics.
We further develop the diffeological space theory from several aspects: categorical, topological and differential geometrical. Our main concern is to build a suitable homotopy theory (also called a model category structure) on the category of diffeological spaces, which encodes the usual smooth homotopy theory of smooth manifolds and the diffeological bundle theory of Iglesias.
This is a huge task, and at the moment, we have not yet completely proved the existence theorem. However, in the process, we can see the beauty of the merging of differential geometry and homotopy theory. (More details are explained in the Introduction.) These results should be of some interest to people working in these fields.
Wu, Enxin, "A Homotopy Theory for Diffeological Spaces" (2012). Electronic Thesis and Dissertation Repository. Paper 661.