The Atiyah-Singer Index theorem is one of the most beautiful results in mathematics. The Index theorem exhibits a rather mysterious relationship between geometric and analytic invariants on manifolds. It is quite remarkable here, that those invariants are totally different in nature. This property makes the Index theorem a very useful tool in many geometric problems. For instance, the integrability of L-genus or A^-genus for closed manifolds is a highly nontrivial consequence.
Recently, exploiting the Index theorem in an elaborate way, several remarkable results in Low-dimensional Topology were obtained by Casson, Donaldson, Floer, Witten and many other people. One of the most remarkable results is due to Taubes, who established that half of the Euler number of Floer homology is nothing but the Cassion invariant for homology spheres. A refined index theorem with respect to the spectral flow played an important role there. Thus, there is no objection to the significance of the Index theorem in geometry, and it is definitely one of the central themes in present day mathematics.
On the other hand Connes has proposed a new framework, the so called Non-commutative Geometry. It is a magnificent proposal spreading to many areas of Mathematics such as Differential Geometry, Algebraic Topology, Operator Algebras, Ergodic Theory, Number Theory, Global Analysis and Mathematical Physics. This is a promising proposal for which we can expect a successful development in the future.
In this COE program our objective is a generalization of the Index theorem in the framework of Non-commutative Geometry, and its applications to geometric problems on manifolds. We shall develop Non-commutative Geometry based on techniques already exploited in the Index theorem, Gauge theory and Integrable systems. When we create non-commutative objects out of commutative ones in whatever context, we encounter geometric discrete objects such as eigenvalues, graphs or groupoids. We shall also employ methods in Graph Theory, Number Theory, Ergodic Theory and Dynamical Systems to establish the notion of a non-commutative manifold.