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I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of various references :
(i) Spin Geometry by Lawson & Michelsohn
(ii) Heat Kernels and Dirac Operators by Berline, Getzler and Vergne
(iii) Dirac operators and spectral geometry by Giampiero Esposito
(iv) Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem by Peter Gilkey
and (v) The laplacian on a Riemannian Manifold by Rosenberg, however I am having difficulty deciding which one or two to study between these. I prefer reading books that start from basics but eventually cover the core aspects of the subject at fairly advanced level. Please advise which one or two of the above should I study such that the intersection of the selected books is minimal and the union is maximal. Any comments about the above-listed books will be very helpful.
My background is : Analysis (As covered in 'Principles of Mathematical Analysis' by Walter Rudin but not much of measure theory), Algbera ( 1 year grad level course based on Serge Lang), Manifold Theory and Differential Geometry (Differential forms, de-Rham theory, Riemannian metrics, Geodesics, Connections, Curvature, Vector bundles & Characteristic classes, Principle bundles ) but very little functional analysis (definition and basic properties of Banach and Hilbert spaces, and the four famous theorems) and almost no Algebraic Topology beyond the definition of fundamental group. By the time I plan to start studying ASIT, I am expecting to have studied Representation theory of Lie groups and Lie Algebras and may be a little bit of Clifford Algebras as well.Please advise what other prerequisites do I need to study the books listed above.In particular, do I need to learn more Functional Analysis and Algebraic Topology for this purpose ? I have studied the prefaces of the books I have listed but unfortunately the information about necessary prerequisites is not mentioned in most cases.
Any other good references are welcome.
Note: I had earlier asked a related question here

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    $\begingroup$ I think you need to take a PDE course. $\endgroup$ – timur Oct 1 '13 at 0:41
  • $\begingroup$ @timur Thanks. Could you please specify which topics in PDE I need to learn and some good text which I could study for this ? $\endgroup$ – user90041 Oct 1 '13 at 14:26
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    $\begingroup$ I think Taylor's second volume would be a good start. $\endgroup$ – timur Oct 2 '13 at 1:10
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    $\begingroup$ What about "Elliptic operators, topology and asymptotic methods" by John Roe $\endgroup$ – Eric O. Korman Oct 3 '13 at 3:33
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    $\begingroup$ You can try my course notes www3.nd.edu/~lnicolae/ind-thm.pdf They're modeled after Roe's book but I go at a slower pace. $\endgroup$ – Liviu Nicolaescu Mar 6 '14 at 12:00
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In have worked for a longer time with the mentioned books of S.Rosenberg (v), P.Gilkey (iv) and Berline, Getzler and Vergne (ii). Since my interests were more related to heat kernels than Dirac Operators I want to comment from this point of view.

The book of S.Rosenberg is excellent to start with. It deals with the basics though trying to touch topics are quite complicated, so it is very good to start diving in.

(ii) is a great source, too. But it's on a more advanced level. I would recommend keeping this book inside your mind but start reading later, if you feel very familiar with the basics - even I can't precisely define, what 'basics' mean.

Finally, (iv) is very often cited directly and belongs to the 'canonical' literature to the topics related to the heat kernel. Definitely worth reading - but don't expect do read it from end-to-end, I would recommend you to pick chapters you are immediately interested in, since it contains a wide range of topics.

edit: If you're seriously working on these topics, you'll be faced with functional analysis automatically, it is an important fundament. I also had no 'further' functional analysis education before, but for me it worked learning parts of it 'on the fly' while considering concrete problems. So don't hesitate if you never heard a special lecture. The book if Rosenberg e.g. does also contain very much of useful functional analytic material.

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  • $\begingroup$ Thanks, this is very helpful. Just one more question : I have had just an undergrad course on Functional Analysis (basic definitions,examples and properties of Banach Spaces, Hilbert spaces, compact operators, and the four famous theorems), can I begin reading Gilkey's book straight away or there are topics in functional analysis which I should learn before starting the book ? $\endgroup$ – user90041 Oct 24 '13 at 3:49
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    $\begingroup$ I think there definitely will be topics you don't know yet, but simply give it a try. Don't expect to read it 'straight'. Start reading a chapter you're curious of and switch to other sources if you start feeling lost. I would recommend you to start with the chapter about Pseudodifferential operators. It's also very useful to compare the mentioned sources while reading. Gilkey mostly handles a more general case of operators, so it is a good idea to compare with methods presented in the book of Rosenberg for the Laplacian on functions. $\endgroup$ – Robin Neumann Oct 24 '13 at 7:33
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I think the best introduction is "Dirac Operators: Yesterday and Today", the proceedings of a summer school edited by Bourguignon, Branson, Chamseddine, Hijazi and Stanton. It does not cover heat kernels, but gives a good understanding of the other topics - and does start from basics; there is a (short) chapter on differentiable manifolds. Also I think a good understanding of the contents here is prerequisite to getting much out of, say, the Lawson-Michelsohn book.

Beyond the prerequisites you mentioned, I think the only thing necessary is a certain understanding of cohomology groups. To start learning about heat kernels, timur is probably right in saying you should first learn some PDE.

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Another worthwhile book that hasn't been mentioned is Liviu Nicolaescu's Lectures on the Geometry of Manifolds which is freely available from his website (under papers'n stuff). While it doesn't cover the Atiyah-Singer Theorem itself, it does a great job of addressing some of the prerequisite material such as elliptic operators (including Dirac operators).

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I have studied the whole book of Rosenberg and I can asure you that it is a very good choice to start with, I have studied some parts of the books of Gilkey and Lawson-Michelson too and I can asure you they are quite advanced, it would be hard to begin with them. I guess the book of Lawson-Michelson is better for you because it covers the classical proof of the Atiyah-Singer index theorem, which depends more on geometry, while Gilkey covers de analytic proof of the theorem, which depends very much on analysis. Anyway, if you prefer to choose the book of Gilkey, be sure to get the second edition, since it is fairly better than the first one. Remark: The book of Gilkey starts in the context of elliptic complexes, which is more general than the context of Dirac operators and Dirac complexes

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