Limiting tangents and normals to singularities of algebraic surfaces (Alan Durfee and Donal O'Shea).
The students were Gautam Chinta (Yale), Melinda Koelling (University of Chicago), Adam Lucas (MIT), Joshua Mandell (Yale), Rachel St. Pierre (Wellesley) and Jeff Zhang (Yale). The group investigated the algebraic and geometric tangent cones and the limit of normals for certain classes of real and complex surface singularities in three-space. In particular, they looked at a family of singularities given by Trotman: zB = xAyT + xA+C.
They calculated the geometric tangent cone (the limit of secants) for all real surfaces in this family by brute force. To find the limit of normals is a more difficult problem. They used a general result of Le and Teissier in the complex case, and almost succeeded in proving it in the real case. They found some general results, and also computed the limit of normals for the members of the Trotman family by brute force. In addition, they worked out two more algebraic methods using Groebner bases. When the surface had a nonisolated singularity they also investigated the relationship between the generic singularity and the special one. For plane curves they completely solved the problem of relating the algebraic and geometric tangent cone; the solution to this problem is probably known to specialists but had never been written down carefully. The group produced a thirty page paper in addition to the pages of computations of special cases.