## [Atiyah–Singer Index Theorem](/atiyah-singer-index-theorem) ## [Atiyah–Singer Index Theorem](https://mathlogic.quantumdictionary.io/atiyah-singer-index-theorem-0) [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic) - Natural & Formal Sciences - **Mathematics & Logic Dictionary** Definition For an elliptic differential operator on a compact smooth manifold, the Atiyah–Singer index theorem states that its analytical index (dimension of kernel minus dimension of cokernel) equals a topological index computable from characteristic classes of the manifold and the symbol of the operator. ## [Atiyah–Singer Index Theorem](https://puremath.quantumdictionary.io/atiyah-singer-index-theorem-1) [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics) - Mathematics & Logic - **Pure Mathematics Dictionary** Definition A deep theorem equating the analytical index of an elliptic differential operator on a compact manifold (the difference of dimensions of kernel and cokernel) with a topologically defined index computed from characteristic classes of the manifold and the operator symbol. ## [Atiyah–Singer Index Theorem](https://algebra.quantumdictionary.io/atiyah-singer-index-theorem-2) [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra) - Pure Mathematics - **Algebra Dictionary** Definition A theorem equating the analytical index of an elliptic differential operator on a compact manifold (the Fredholm index counting kernel and cokernel) with a topological index computed from characteristic classes of the operator's symbol in K-theory, thereby connecting analysis, topology, and algebraic K-theory.