Talk:Atiyah–Singer index theorem

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s is a bundle section and required to be non-zero. E.g. for a Laplacian s is a positive-definite quadratic form.

If the symbol s is a bundle section, then it is a function defined on the manifold M. How could it be a positive-definite quadratic form? M need not be a vector space.

For the Laplacian, the symbol would be p²+q², and in general for constant coefficients one should apply some sort of Fourier transform to turn polynomials in derivatives to ordinary polynomials; and then take the top homogeneous term. Variables like p and q have an intrinsic meaning in the cotangent bundle (I think) - this means that homogeneous polynomials in them exist in some tensor bundle formed by symmetric powers. The important property for being elliptic is that the symbol should have no real zeroes, other than e.g. p=q=0. If one looks only at constant-coefficient operators, that is the same as requiring translation-invariance (which is why the symbol would be the same everywhere on M, since that is being treated as a vector space). So, the symbol in general is going to vary from point to point; ellipticity has to be defined as the condition that it is not zero, except trivially. The general statement has to be quite, err, general since the operator will go from sections of one vector bundle to those of another, and :saying where s lives is already quite complex. Charles Matthews 13:19, 9 Apr 2004 (UTC)

Also, could we have the simplest case of the theorem explained? Like M=S¹ and the simplest elliptic operator on the smooth functions defined on M. AxelBoldt 00:50, 28 Mar 2004 (UTC)

On the circle, I think it should be id/dx (up to sign), and the statement should be the index is 0 in this case. Differentiation has kernel of dimension 1. And the cokernel is also of dimension 1; i.e. F(θ) to be a derivative has the necessary condition to satisfy, that it should integrate to zero around the circle, and this is also sufficient. So topologically this would correspond to the Euler characteristic of the circle being 0, also.

Charles Matthews 13:19, 9 Apr 2004 (UTC)

Formal version?

While searching for a formal statement of the theorem, I came across this paper [1] on Google. It has a theorem on page 3 that looks like it *might* be the Atiyah-Singer theorem, but I don't know enough about these things to determine whether it really is or not. Can someone look into this, maybe? Thanks. -- Schnee 00:05, 30 Oct 2004 (UTC)

Clearly closely related, but it's a generation more advanced in what it is trying to do. There would be even more of a problem in writing that here, of too many undefined terms. Charles Matthews 09:07, 30 Oct 2004 (UTC)

Ah, OK. I probably should try to see if my hometown's university's mathematical library has the journal that Atiyah and Singer published their results in, but I don't know when I'll be there again. I'll add the references to the article, though. -- Schnee 15:32, 30 Oct 2004 (UTC)

BTW, there's also an interesting book (in Postscript format) at [2] titled "The Atiyah-Patodi-Singer Index Theorem" (a generalization of the Atiyah-Singer index theorem as far as I understand). Again, I don't know enough about this to make much use of it, but maybe someone else can do. -- Schnee 00:10, 30 Oct 2004 (UTC)

Applications in Microeconomics?

It seems to me that the "index theorem" used in General Equilibrium Theory is actually the Poicare-Hopf theorem and not this one. Any comments?

I have deleted "applications to general equilibrium" until we find a reference. --Rastaco 13:25, 17 February 2006 (UTC)[reply]

Topological index for dummies?

Can someone please give an explanation of the topological index for dummies? The analytical index is intuitive but the topological index is some complicated unintuitive mess. The literature isn't all that helpful. All it does is to give one complicated formal definition after another without explaining just what the hell it is all about. Todd class, Thom isomorphism, Chern class, etc.

theoretical physics applications

aren't mentioned --euyyn 08:20, 23 April 2007 (UTC)[reply]

Is the definition of "elliptic operator" correct?

The article says:

"The operator is called elliptic if the symbol is nonzero whenever at least one y is nonzero.

Example: The Laplace operator in k variables has symbol y12 + ... + yk2, and so is elliptic as this is nonzero whenever all the y's are nonzero. The wave operator has symbol −y12 + ... + yk2, which is not elliptic if k ≥ 2, as the symbol vanishes for some non-zero values of the ys."

It should probably say: "The operator is called elliptic if the symbol is zero iff all the ys are zero."

but there is a positivity condition as well, isn't there?

As it stands, the wave operator example does not follow from the definition at all (the definition says nothing about the zero case, only about the nonzero case) and the word "all" in the Laplace case is wrong: it should be "any", but the whole sentence needs to be rewritten so that it follows from the correct definition. —Preceding unsigned comment added by Ndokos (talk • contribs) 12:55, 10 October 2007 (UTC)[reply]

add a reference?

I'd like to add an important reference for the "difference element" defined in K-theory (appearing in the definition of the topological index), but I'm afraid of doing it wrong.

The reference is the 1961 paper of Atiyah and Hirzebruch, "Analytic cycles on complex manifolds", section 3. — Preceding unsigned comment added by 129.49.88.221 (talk) 19:59, 10 April 2014 (UTC)[reply]

Computations Needed

This page should include some explicit computations of the index theorem to give a better overview for how the index theorem works. In addition, the index theorem should be presented in the form given here: https://math.stackexchange.com/a/252225 I think this is much more slick of an explanation. — Preceding unsigned comment added by 128.138.65.80 (talk) 22:08, 4 December 2017 (UTC)[reply]

There is a great set of lecture notes which give the tools necessary for performing these computations: https://web.archive.org/web/20130124054444/http://www.indiana.edu/~jfdavis/teaching/m721/mano.pdf — Preceding unsigned comment added by 50.246.213.170 (talk) 23:10, 26 July 2018 (UTC)[reply]

Notation for Manifolds

It's common to use X, Y for topological spaces and M, N for Manifolds. But in the article X, Y has been used for manifolds which is confusing. I wanted to change it but a picture has been used with X, Y. So I couldn't change it right now. Maybe in future (Or if anybody else could change it would be good thx) Mojtabakd «talk» 09:00, 10 June 2021 (UTC)[reply]

This sounds like a good idea to me, but what do I know. By picture, I guess you mean the commutative diagram. Here try this:

{\begin{array}{ccc}&&&\\&K(M)&{\xrightarrow[{}]{{\text{Td}}(M)\cdot {\text{ch}}}}&H(M;\mathbb {Q} )&\\&f_{*}{\Bigg \downarrow }&&{\Bigg \downarrow }f_{*}\\&K(N)&{\xrightarrow[{{\text{Td}}(N)\cdot {\text{ch}}}]{}}&H(N;\mathbb {Q} )&\\&&&\\\end{array}}

Seems to work. 67.198.37.16 (talk) 04:22, 30 May 2024 (UTC)[reply]