Atiyah?Singer index theorem

In the mathematics of manifolds and differential operators, the Atiyah?Singer index theorem is an important unifying result that connects topology and analysis. It deals with elliptic differential operators (such as the Laplacian) on compact manifolds. It finds numerous applications, including many in theoretical physics and equilibrium theory in microeconomics.

Related Topics:
Mathematics - Manifold - Differential operator - Topology - Analysis - Elliptic differential operator - Laplacian - Compact - Theoretical physics - Microeconomics

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When Michael Atiyah and Isadore Singer were awarded the Abel Prize by the Norwegian Academy of Science and Letters in 2004, the prize announcement explained the Atiyah?Singer index theorem in these words:

Related Topics:
Michael Atiyah - Isadore Singer - Abel Prize - Norwegian Academy of Science and Letters - 2004

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Scientists describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas, called differential equations, involving their rates of change. Such formulas may have an "index," the number of solutions of the formulas minus the number of restrictions that they impose on the values of the quantities being computed. The Atiyah?Singer index theorem calculated this number in terms of the geometry of the surrounding space.

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A simple case is illustrated by a famous paradoxical etching of M. C. Escher, "Ascending and Descending," where the people, going uphill all the time, still manage to circle the castle courtyard. The index theorem would have told them this was impossible.

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