Few theorems can be said to have opened up entire new areas of mathematical research - but this is what the Riemann Roch theorem managed.

Initially considered by Riemann as a helpful way of approximating the dimension of the space of certain meromorphic functions on a Riemann surface; this was then refined by his student Roch to give a precise formula for the dimension.

The applications of this single formula are numerous; but this formula also lead to many deep generalisations, starting with Hirzebruch's interpretation of the theorem in terms of vector bundles (and his use of algebraic topology, in particular the theory of characteristic classes, to express this). A few years later followed Grothendieck's interpretation of the theorem, changing the absolute viewpoint to a relative one, something new for the time, and introducing ideas that were going to lead to K-Theory.

Meet us in MS.05 at 7:30pm for an introduction to Riemann surfaces and the Riemann-Roch Theorem (not the generalizations!). Then we invert some line bundles and appear in the pub.