Riemann-Roch Theorem/Motivation

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Motivation for Riemann-Roch Theorem

The Riemann-Roch Theorem gives a formula which is used to calculate the exact number of independent holomorphic functions that can be defined on a given Riemann surface.

This formula involves the number of independent differential forms of special kinds.

Some of these are equal to topological invariants of the Riemann surface.

These are, in general, easy to calculate.


Historical Note

The Riemann-Roch theorem was stated and proved in the mid-$19$th century by Bernhard Riemann and his student at the time Gustav Roch.

Bernhard Riemann supposedly achieved his version by considering configurations of closed spaces and imagining experiments performed on them.

It has been used in recent times in the study of codes.


Sources