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Atiyah-Singer Theorem: Difference between revisions
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In differential geometry, the AtiyahâSinger index theorem 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). It includes many other theorems, such as the ChernâGaussâBonnet theorem and RiemannâRoch theorem, as special cases, and has applications to theoretical physics. | In differential geometry, the AtiyahâSinger index theorem 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). It includes many other theorems, such as the ChernâGaussâBonnet theorem and RiemannâRoch theorem, as special cases, and has applications to theoretical physics. | ||
: | : <math>dim\, ker \not{D}_E - dim \, coker \not{D}_E = \int_M \hat{A}(M) \cdot ch(E)</math> | ||
== Resources: == | == Resources: == | ||
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*[https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem Atiyah-Singer Theorem] | *[https://en.wikipedia.org/wiki/Atiyah%E2%80%93Singer_index_theorem Atiyah-Singer Theorem] | ||
== | == Related Pages == | ||
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* [[Isadore Singer]] | |||
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