Normal forms are useful in differential geometry.
The Moser normal form of a CR hypersurface is a good example. Affine differential geometry in its simplest guise is the study of those geometric properties of submanifolds in R n invariant under affine transformations. A hypersurface which is the orbit of a Lie subgroup of the group of affine motions is said to be affine homogeneous. The Archimedean screw in R 3 is a good example as are ellipsoids and the parabolic dish. This talk will discuss normal forms and how they may be used to define invariants and classify homogeneous surfaces in R 3 and more besides.