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Do you know a basic reference to introduce an undergraduate student with more physical rather than mathematical background to De Rham cohomology?

The Student (from a Bachelors program in material science) should work on a summer project aiming to understand some of the mathematics behind the quantum hall efect, explained with chern numbers. The student has previously helped on experiments with some materials relevant to the quantum hall efect.

I do know the mathematical ellaborations on this ranging von Bellisard, Schulz-Baldes, collaborators and Connes'approach on the topic in the noncommutative geometry book. This will be certanily too advanced from the point of view of mathematics.

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  • $\begingroup$ What about an elementary differential geometry textbook? (I am thinking of a mixture of chapters from Lee's differentiable manifolds plus the appendix of Milnor-Stasheff for the characteristic classes material). Would that also be too advanced? $\endgroup$
    – Denis Nardin
    Commented May 16, 2017 at 16:32
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    $\begingroup$ I like the book "From Calculus to Cohomology" by Madsen and Tornehave for this sort of thing, as well as the "Topology, Geometry, and Gauge Fields" series by Naber. But starting with nothing and getting through de Rham cohomology in one summer feels ambitious, let alone the quantum hall effect. $\endgroup$
    – Paul Siegel
    Commented May 16, 2017 at 17:11
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    $\begingroup$ Why is this appropriate on MO rather than on MSE? $\endgroup$
    – Alex M.
    Commented May 16, 2017 at 17:41

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If you are familiar with some topological field theory, then the de Rham cohomology of $M$ can be viewed as the vector space of supersymmetric vacua in the $\sigma$-model associated to $M$. I think this was first noted by Prof. Witten in his work on sypersymmetry and Morse theory. You can look it up in the paragraph 4 of chapter 10 here.

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I think the book you want is Frankel's The Geometry of Physics: An Introduction. It's accessible and well written, and largely aimed at the material you're talking about.

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