A deformation of the cubical Fermat-Pham-Brieskorn function on .

The cubical Fermat-Pham-Brieskorn function is given by the expression . The mapping , given by , is a continuous bijection. The reciprocal map is also continuous and is given by . Please notice that the reciprocal map is not differentiable. It follows that maps the level surface to the plane . Hence the level surfaces , are topological planes in . Since the reciprocal map is not differentiable, we can not conclude, that all the level surfaces , are differentiable submanifolds in . The level surfaces of the function produce a non differentiable foliation on , which is topologically the standard foliation of by planes,

The function
is a deformation of the function . The functions
have critical points, which are
the vertices
of a cube in
.
The maximal number of critical points, which can have a function that is a
``small'' perturbution of the function , is the sum of the
Milnor numbers of its singularities. The function
has only a singularity at with
Milnor number . The deformation
is
called a *maximal small deformation* of the singularity
of the function at
, since this deformation realizes
the maximal possible number of critical points. Please
notice that for negative values of the function
has no critical points at all.

The movie SURFACE shows level surfaces of the function . One can see a ballet of appearing and disappearing cycles. From the choreography and geometry of this ballet one can determine the toplogical behavior of the corresponding complex map .

The frames of the movie are made with the software SURF, which is freely available at http://surf.sourceforge.net/. The playoff is done with XANIM, with is part of any Linux distribution.