This award supports preliminary research on a topological invariant, -P.P, associated to the germ of any complex normal surface singularity. The goal is to develop techniques for both finding a volume form on the link of a surface singularity, and for formulating closed expressions for -P.P. The principal investigator will initially work on determining the behavior of -P.P under splicing and finite ramified maps. This is research in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.
