Noncommutative Geometry: New Frontiers

A remarkable development in modern physics has been the recognition that geometry and topology play an important role in fundamental theories of matter such as quantum mechanics. This has only been elucidated in a piecemeal way over decades but recent revolutionary work of Alain Connes gives a new mathematical theory, noncommutative geometry ideally suited to quantum theory. Quite dramatic new applications have been found in both mathematics and physics.This project will resolve fundamental questions in Connes theory. It will broaden its range of applicability, generate new approaches to the mathematics of index theory and provide new tools in quantum theory.