Written Qualifier Exam

By “generic” we mean stable singularities (intersections), which cannot be reduced by adding small perturbations to the functions/data. That is, what kind and how many configurations are possible such that small changes in the shape of regions near the intersection point do not/cannot reduce the number of regions meeting at a single point. The term “small changes” is somewhat ambiguous in this description, but can be taken to mean infentesimally small deformations. In 2D, there are three types of intersections, 1 region, 2 regions, or three regions. Figure 1 A shows the region configurations for two, three, and four material intersections. Only the two and three intersection configurations are stable/generic. The four material intersection is easly perturbed into two three-material intersections; this is accomplished by moving the top two material intersection slightly left or right. Therefore, there are only two generic types of singularities for 2D regions, the two-region intersection, sometimes refered to as a fold, and the three-region intersection, sometimes refered to as a cusp.