# Geometry of Algorithms

The study of properties of algorithms invariant under a transformation needs a name. How about Geometry of Algorithms? Projective Geometry of Algorithms will study properties invariant under projective transformations like computer programs, simplicial chain embeddings, etc.

An algorithm performs a specific function. Each algorithm has many programs in a specific language. The language offers the instructions, while the programs specify the incidence relations between inputs and instructions. The projective geometry of an algorithm will provide a framework to embed a mathematical theorem into a program, like how stochastic calculus helps embed discrete algorithms into continuous functions.

What defines an algorithm's homology? Can one program probe another geometrically? Does this relate to algorithm convolution? These questions seem nonsensical until you realize billions of algorithms are rehashes of a few hundred geometric ones like FFT, Euclidean algorithm, Quicksort, Subspace Iteration, Fast Multipole, etc. These form the geometric basis for others. Viewing incidence relations as complex building blocks, an algorithm's geometry is an n-dimensional chain complex. These complexes can be convolved for interesting results. Algorithms with similar functions share similar geometry. An algorithm true to its input/output should behave the same in geometric space, simplifying proof management systems.