The Atiyah-Singer Index Theorem relates the analytical index of an elliptic differential operator on a compact manifold, defined by counting solutions to equations associated with the operator, to purely topological data of the manifold and the operator's own symbol. Proved by Michael Atiyah and Isadore Singer, it unifies results from analysis, topology and geometry and has become a central tool connecting those fields.
Facts
StatementFor an elliptic differential operator on a compact manifold, the analytical index, which counts solutions to equations associated with the operator, equals the topological index, a quantity defined purely from topological data of the manifold and the operator's own symbol. 1 Connections
Sources
1. Atiyah-Singer Index Theorem (Wikipedia)
Wikimedia FoundationLead section, statement of the theorem
the Atiyah-Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data).
History section, on the 1963 announcement
The Atiyah-Singer theorem was announced in 1963.
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