In 40's Whitney studied maps of C^\infty manifolds. When a map is not an
immersion/submersion, one tries to deform it locally, in hope to make it 'generic'.
This approach has led to the rich theory of stable maps, developed by Mather, Thom and
many others.
The main 'engine' was vector field integration. This chained the whole theory to the
C^\infty, or R/Canalytic setting. I will present the purely algebraic approach,
studying maps of germs of Noetherian schemes, in any characteristic. The relevant
groups of equivalence admit 'good' tangent spaces. Submodules of the tangent spaces
lead to submodules of the group orbits. Then goes the theory of unfoldings (triviality
and versality). Then I will discuss the new results on stable maps and theorems of
MatherYau/GaffneyHauser.
