Abstract
The Atiyah–Singer index theorem has been vastly generalized to higher index theory for elliptic operators in the context of noncommutative geometry. Higher index theory has important applications to problems in differential topology and differential geometry such as the Novikov Conjecture on homotopy invariance of higher signatures and the existence problem of Riemannian metrics with positive scalar curvature. In this article, I will give a survey on recent development of higher index theory, its applications, and its fascinating connection to the geometry of groups and metric spaces.
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