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I would like to know what a Penrose transform is, or more precisely, what is it intended to be - I'm interested in ideas, intuition and some examples of application. My knowledge of differential geometry and homological algebra are basic (at the moment, I am working on a graduation paper on derived categories), so please be rather broader than technically specific in the answer if possible.

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    $\begingroup$ It might help to say why the Wikipedia page doesn't do what you want. $\endgroup$
    – Mark Grant
    Sep 7, 2014 at 11:25
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    $\begingroup$ this thesis in particular might get you a long way --- maths.ed.ac.uk/pg/thesis/david.pdf $\endgroup$ Sep 7, 2014 at 11:44
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    $\begingroup$ @MarkGrant the Wikipedia page says that Penrose tranform relates massless fields on spacetime to cohomology of sheaves on complex projective space, but doesn't say anything about why one would be interested in such a relation. Is it computation, theoretical viewpoint or something else? To be blunt, what does exactly Penrose transform achieve that is hard to do otherwise? Wikipedia continues with saying that inspection of cohomology classes yields massless field equations for a given spin. Is this something we wanted to calculate or just confirms the value of the concept? $\endgroup$
    – Ennar
    Sep 7, 2014 at 14:13
  • $\begingroup$ @CarloBeenakker Thank you for the reference! $\endgroup$
    – Ennar
    Sep 7, 2014 at 14:14

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well, if you are interested in developing some basic intuition on the Penrose transform, you could go back to Harry Bateman's 1904 paper The solution of partial differential equations by means of definite integrals, where he derived what is essentially the same representation of a harmonic function $\phi(w,x,y,z)$ of four variables as a contour integral of a holomorphic function $f(p,q,\zeta)$ of three complex variables:

$$\phi(w,x,y,z)=\oint f[w+ix+(iy+z)\zeta,iy-z+(w-ix)\zeta,\zeta]\,d\zeta$$

An instructive geometric interpretation of Bateman's formula, which connects it to Penrose's work, is given by Michael Eastwood in his "Introduction to Penrose Transform". (You can read most of it at books.google.com and find a brief summary here.) It is also interesting to read Roger Penrose's own account of the relation between Bateman's formula and the Penrose transform.

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  • $\begingroup$ Thank you for these references, I guess it will serve more than enough as an introduction. $\endgroup$
    – Ennar
    Sep 23, 2014 at 13:30

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