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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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V.2 The Atiyah–Singer Index Theorem

Nigel Higson and John Roe

1 Elliptic Equations

The Atiyah–Singer index theorem is concerned with the existence and uniqueness of solutions to linear partial differential equations of elliptic type. To understand this concept, consider the two equations

Image

They differ only by the factor i = image, but their solutions nevertheless have very different properties. Any function of the form f (x, y) = g(x - y) is a solution to the first equation, but in the analogous general solution g(x + iy) of the second equation, g must ...

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