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.



Idea and intuition behind Penrose transform
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.](http://www.msri.org/ext/concepts/eastwood3.pdf)) It is also interesting to read Roger Penrose's [own account](http://users.ox.ac.uk/~tweb/00001/) of the relation between Bateman's formula and the Penrose transform.


