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The present publication contains a special collection of research and review articles on deformations of surface singularities, that put together serve as an introductory survey of results and methods of the theory, as well as open problems, important examples and connections to other areas of mathematics. The aim is to collect material that will help mathematicians already working or wishing to work in this area to deepen their insight and eliminate the technical barriers in this learning process. This also is supported by review articles providing some global picture and an abundance of examples. Additionally, we introduce some material which emphasizes the newly found relationship with the theory of Stein fillings and symplectic geometry. This links two main theories of mathematics: low dimensional topology and algebraic geometry.
The theory of normal surface singularities is a distinguished part of analytic or algebraic geometry with several important results, its own technical machinery, and several open problems. Recently several connections were established with low dimensional topology, symplectic geometry and theory of Stein fillings. This created an intense mathematical activity with spectacular bridges between the two areas. The theory of deformation of singularities is the key object in these connections.
Content Level »Research
Keywords »algebraic geometry - low dimensional topology - singularity theory
Altmann, K. and Kastner, L.: Negative Deformations of Toric Singularities that are Smooth in Codimension Two.- Bhupal, M. and Stipsicz, A.I.: Smoothing of Singularities and Symplectic Topology.- Ilten, N.O.: Calculating Milnor Numbers and Versal Component Dimensions from P-Resolution Fans.- Némethi, A: Some Meeting Points of Singularity Theory and Low Dimensional Topology.- Stevens, J.: The Versal Deformation of Cyclic Quotient Singularities.- Stevens, J.: Computing Versal Deformations of Singularities with Hauser's Algorithm.- Van Straten, D.: Tree Singularities: Limits, Series and Stability.