Novel geometric constructions

The topology and geometry of higher dimensional spaces remains a core area of high level research activity in mathematics. When these spaces are not completely smooth but have singularities or singular subspaces there is a great paucity of results that use index theory. By contrast generalisations of the Atiyah-Singer index theorem provide some of the major tools in the smooth case. This proposal addresses this deficiency by developing new geometric constructions and invariants for singular spaces. The outcomes will resolve questions in geometry and topology that have arisen as a result of recent breakthroughs in index theory and higher structures in differential geometry.