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Mathematics Colloquium

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Date:  Tuesday, February 14, 2006

Title:  Topology and Equations of Complex Surface Singularities

Abstract:  This talk concerns the relationship betwen the topology near a singular point of a complex surface, and the equations which define the singularity. The vanishing of a complex polynomial in 3 variables specifies a complex surface in 3-space; at a singular point (one where all derivatives vanish), the local topology is given by intersecting the complex surface with a small 5-sphere centered at the singular point, giving a compact connected 3-manifold called the "link" of the singularity. Milnor and Mumford did fundamental work in the 1960's and we now understand which 3-manifolds can occur, but it is hard in general to go from information about the link to information about the defining equations. Much progress has been made by work of M. Artin, Laufer, Dolgachev, Pinkham, Neumann, and others. In the talk we first discuss the case of singular points of plane curves, where the link is a knot in 3-space. Then we describe results of joint work with Walter Neumann on links for complex surfaces. A motivating example for our work is Felix Klein's calculations of the ring of invariants of a finite subgroup of SU(2) acting on the ring of polynomials in two variables--one should look at the universal abelian covering of this singularity.


Speaker:  Jonathan Wahl
Institution:  University of North Carolina

 

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