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 and
examples. 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. 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.
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