Singularities JPGs

The selections below will display JPG versions of the twelve illustrations from my book, Le (L\^e) Cycles and Hypersurface Singularities, which appears in Springer-Verlag's Lecture Notes in Mathematics series, volume 1615 (1995).

All of these images were created using Mathematica and ClarisDraw. However, since I have converted them to JPGs here, they do not look nearly as good as the postscript versions.

• An illustration showing the Milnor fibration inside a closed ball.

• The coordinate hyperplanes in C^3 (or R^3).

• A picture of the Whitney umbrella in R^3, including the handle!

• Nodes degenerating to a cusp, with the L\^e and relative polar cycles sketched in.

• The hypersurface defined by h = z^2 + (x - y^2)^2.

• A generalization of the example of nodes degenerating to a cusp.

• A picture of the critical locus of h = y^2 - x^3 - (u^2 + v^2 + w^2)x^2 ; an example which I usually refer to as the FM cone.

• A technical diagram needed in a homotopy argument.

• Side-by-side pictures of a one-dimensional singularity and the isolated singularity produced from it by adding a large power of a generic linear form.

• A shot of the famous swallowtail singularity from a nonstandard point of view. This viewpoint is intended to be a good one from which to see the complicated activity at the origin.

• An illustration of the failure of Thom's a_f condition along an axis. Shown are two level surfaces of a function f , together with tangent planes. These planes are approaching a plane at the origin which does not contain the tangent plane to the indicated axis at the origin.

• Another diagram needed in a homotopy argument. Over the five fifth roots of unity, there are contractible fibres. Over the other points in the disc, the fibres are not contractible.