Some things that I really would like to make good drawings of
Very bad grammar in the title, sorry.
The ``things that I really would like to make good drawings of'' are unfoldings of two codimension- singularities in the stratified space of smooth functions from
to
. The two are the the butterfly singularity, or hyperbolic umbilic, and the elliptic umbilic, or monkey saddle. In fact, the butterfly singularity lies in the space of functions from
to
and the monkey saddle in the space of functions from
to
, but by adding sums of squares in the other coordinates, these give codimension-
singularities for maps from
to
. Incidentally, the best pictures I know of in print are in Hatcher and Wagoner's ``Pseudo-isotopies of compact manifolds'', but we should be able to do better now with our increased computational power.
The butterfly singularity is, well, pretty simple: . Seems strange to give it such a special name, but this becomes more meaningful when you think about its ``universal unfolding''. This function can of course be perturbed in many different ways to make it Morse. The ``universal unfolding'' is
. For generic values of
,
and
this will be a Morse function, and as we move around in
parameter space we see the singularities coming together and cancelling in various ways. What I want is a nice way to draw, for a given plane
in
-space, the immersed surface
in
defined by
if
for some
with
. This shouldn't be too hard, right? Then I want to animate this as we move this plane
through
-space.
The monkey saddle is ... arggh, lost some text here when editing this post later... moral: don't edit your posts after you've written and posted them... will try to fix it later...