Talk:Atiyah–Singer index theorem

message: :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)
thread_title: Untitled

message: While searching for a formal statement of the theorem, I came across this paper  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)
replies: text: 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)
replies: text: 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)
thread_title:  Formal version? 

message: 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)
thread_title:  Applications in Microeconomics? 

message: R.e.b: Since you're immersed in a big revision of this page, I don't want to interrupt with anything other than trivial edits, but while searching for something unrelated to the ASIT, I came across another possible external link, namely Martin Bendersky's lecture notes. I will leave it up to you to decide if it is worthwhile. Michael Kinyon 17:58, 29 September 2006 (UTC)
replies: text: It talks about the Hödge theorem! One of the all-time great typos. Charles Matthews 19:56, 29 September 2006 (UTC)
thread_title: Other external link?

message: aren't mentioned --euyyn 08:20, 23 April 2007 (UTC)
thread_title:  theoretical physics applications 

message: 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)
thread_title:  Is the definition of "elliptic operator" correct? 

message: 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)
thread_title:  add a reference? 

message: 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)
thread_title:  Computations Needed