Atiyah-Singer Theorem: Difference between revisions
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'''Michael Atiyah'''(b. 1929)<br> | '''Michael Atiyah''' (b. 1929)<br> | ||
'''Isadore Singer'''(b. 1924) | '''Isadore Singer''' (b. 1924) | ||
'''''Atiyah–Singer index theorem''''' 1963 | '''''Atiyah–Singer index theorem''''' 1963 | ||
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In differential geometry, the Atiyah–Singer 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 Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics. | In differential geometry, the Atiyah–Singer 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 Chern–Gauss–Bonnet theorem and Riemann–Roch 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 == | ||
* [[Isadore Singer]] | |||
[[Category:Pages for Merging]] | |||
Latest revision as of 20:28, 24 November 2025
Michael Atiyah (b. 1929)
Isadore Singer (b. 1924)
Atiyah–Singer index theorem 1963
In differential geometry, the Atiyah–Singer 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 Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics.
- [math]\displaystyle{ dim\, ker \not{D}_E - dim \, coker \not{D}_E = \int_M \hat{A}(M) \cdot ch(E) }[/math]