  Many-fermion Hilbert space has the algebraic structure of a free module
generated by a finite number of antisymmetric functions called shapes.
Physically, each shape is a many-body vacuum, whose excitations are described
by symmetric functions (bosons). The infinity of bosonic excitations accounts
for the infinity of Hilbert space, while all shapes can be generated
algorithmically in closed form. The shapes are geometric objects in
wave-function space, such that any given many-body vacuum is their
intersection. Correlation effects in laboratory space are geometric constraints
in wave-function space. Algebraic geometry is the natural mathematical
framework for the particle picture of quantum mechanics. Simple examples of
this scheme are given, and the current state of the art in generating shapes is
described from the viewpoint of treating very large function spaces.
