Conversely the index theorem for an elliptic complex can easily be reduced to the case of an elliptic operator: By taking Y to be some sphere that X embeds in, this reduces the index theorem to the case of spheres.
If there is a group action of a group G on the compact manifold X, commuting with the elliptic operator, then one replaces ordinary K-theory with equivariant K-theory.
In this case the index is an element of the K-theory of Y, rather than an integer. These have an asymptotic expansion for small positive t, which can be used to evaluate the limit as t tends to 0, giving a proof of the Atiyah—Singer index theorem.
In Constantine Calliasat the suggestion of his Ph. If Y is a sphere and X is some point embedded in Y, then any elliptic operator on Y is the image under i! These cancellations were later explained using supersymmetry. Most version of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators.
If the manifold is allowed to have boundary, then some restrictions must be put on the domain of the elliptic operator in order to ensure a finite index.
For example, a pseudoinverse of an elliptic differential operator of positive order is not a differential operator, but is a pseudodifferential operator.
This gives a little extra information, as the map from the real K-theory of Y to the complex K-theory is not always injective.
The Atiyah—Singer index is only defined on compact spaces, and vanishes when their dimension is odd. So the index theorem can be proved by checking it on these particularly simple cases.
In this case, constant coefficient differential operators are just the Fourier transforms of multiplication by polynomials, and constant coefficient pseudodifferential operators are just the Fourier transforms of multiplication by more general functions. In fact, for technical reasons most of the early proofs worked with pseudodifferential rather than differential operators: Many proofs of the index theorem use pseudodifferential operators rather than differential operators.
The idea of this first proof is roughly as follows. Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space Y. Consider the ring generated by pairs X, V where V is a smooth vector bundle on the compact smooth oriented manifold X, with relations that the sum and product of the ring on these generators are given by disjoint union and product of manifolds with the obvious operations on the vector bundlesand any boundary of a manifold with vector bundle is 0.
Then one checks that these two functions are in fact both ring homomorphisms. This is similar to the cobordism ring of oriented manifolds, except that the manifolds also have a vector bundle.
Atiyah showed how to extend the index theorem to some non-compact manifolds, acted on by a discrete group with compact quotient.Singular spectrum analysis [4,7,8,13,15,34,35] is a powerful method of time series analysis, which does not require a parametric model of the time series given in advance and therefore SSA is very well suitable for exploratory analysis.
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In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (), 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 Riemann–Roch.Download