It seems to me that adding the word theory tends to increase my interest in a subject. Consider puddles, they’re fun…but ‘Puddle Theory’ what’s that? Sounds intriguing.
Definition: An n-puddle is a n-dimensional manifold with boundary such that the boundary is the union of a sky and ground. The ground is a n-1 manifold with boundary equal to the n-2 boundary of the sky. The sky boundary resides within a hyperplane of dimension n-1.
We can use standard geometric structures on because in general an n-puddle will not be smooth.
Definition: Define a splash to be a family of functions indexed by . need not be continuous, in fact will seldom be so.
It’s possible for to become disconnected during the splash. Splashes can be divided into volume-preserving and non-volume-preserving splashes. These are called dry and wet respectably.
Definition: A splash is dry if .
It’s possible to compose a series of splashes, such that . It was discovered by A. Child that for any such sequence there exists an integer N such that for all is the empty splash. This is known as Child’s Theorem. We leave this as an exercise.