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Atiyah-Singer index theorem
In the mathematics of manifolds and differential operators, the Atiyah-Singer index theorem is an important unifying result that connects topology and analysis. It deals with elliptic differential operators (such as the Laplacian) on compact manifolds. It finds numerous applications, including many in theoretical physics and equilibrium theory in microeconomics.
When Michael Atiyah and Isadore Singer were awarded the Abel Prize by the Norwegian Academy of Science and Letters in 2004, the prize announcement explained the Atiyah-Singer index theorem in these words:
Scientists describe the world by measuring quantities and forces that vary over time and space. The rules of nature are often expressed by formulas, called differential equations, involving their rates of change. Such formulas may have an "index," the number of solutions of the formulas minus the number of restrictions that they impose on the values of the quantities being computed. The Atiyah-Singer index theorem calculated this number in terms of the geometry of the surrounding space.
A simple case is illustrated by a famous paradoxical etching of M. C. Escher, "Ascending and Descending," where the people, going uphill all the time, still manage to circle the castle courtyard. The index theorem would have told them this was impossible.
More formal statement
We start with a compact smooth manifold M (without boundary), a vector bundle on M and an elliptic operator E on M. Here E is a differential operator acting on smooth sections of the vector bundle. The property of being elliptic is expressed by a symbol s that can be seen as coming from the coefficients of the highest order part of E; s is a bundle section and required to be non-zero. E.g. for a Laplacian s is a positive-definite quadratic form.
By some basic analytic theory the differential operator E gives rise to a Fredholm operator. Such a Fredholm operator has an index, defined as the difference between the dimension of the kernel of E (solutions of Ef = 0, the harmonic functions in a general sense) and the dimension of the cokernel of E (the constraints on the right-hand-side of an inhomogeneous equation like Ef = g).
The index problem is the following: compute the index of E using only the symbol s and topological data derived from the manifold and the vector bundle. The Index Theorem solves this problem. Its precise statement requires K-theory, as well as a background in functional analysis and pseudo-differential operators in the manifold setting (sometimes called global analysis).
The theorem came at the end of more than 100 years' development on the theory of elliptic operators (such as Laplacians), going back to the Riemann-Roch theorem. The index problem may have been posed in generality first in the late 1950s by Israel Gel'fand. Given the examples from Hodge theory, Cauchy-Riemann operators in several variables, and the topologists' work on the Riemann-Roch Theorem at the time, the required concepts were perhaps all 'up in the air' by 1960.
In papers written or published in the period around 1962-1965 the theorem was stated and proved by Michael Atiyah, Raoul Bott and Isadore Singer. The proof required (in effect) the rediscovery of the Dirac equation, and the use of complexes of operators.
Atiyah promoted for a while a notion of elliptic topology for which the index theorem was the central notion. Applications were found on a broad front, for example to fixed-point theory, and the representations of Lie groups.
In a further wave of development, the heat equation was introduced to re-prove the Index Theorem. This became a more standard analytical approach, removing perhaps some of the apparent depth of the theory.
There have been a number of subsequent developments, in particular in the work of Alain Connes.
- Peter B. Gilkey: Invariance Theory, the Heat Equation, and the Atiyah-Singer Theorem. Free online textbook that proves the Atiyah-Singer theorem with a heat equation approach
- Richard B. Melrose: The Atiyah-Patodi-Singer Index Theorem. Free online textbook.
- Atiyah, Michael F. and Singer, Isadore M., The Index of Elliptic Operators on Compact Manifolds, Bull. Amer. Math. Soc. 69, 322-433, 1963.
- Atiyah, Michael F. and Singer, Isadore M., The Index of Elliptic Operators I, II, III. Ann. Math. 87, 484-604, 1968.
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