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The topology of positive scalar curvature



Abstract
Given a smooth compact manifold M without boundary, does it admit a Riemannian metric with posi- tive scalar curvature everywhere? This question reveals deep connections between geometry, topology, and analysis (through the modern method to answer it). The most classical answer is given for 2-dimensional surfaces by the Gauss-Bonnet theorem: if a manifold has positive curvature then its Euler characteristic is positive. In higher dimensions, the role of the Euler characterestic is taken by the index of the Dirac operator. To make ecient use of this, recent developments of operator algebras have to be used. The talk will mainly try to introduce the basics of these exciting methods, and explain in examples how it works.