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The objective of this book is to give a comprehensive presentation of mathematical constructions and tools that can be used to study problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. In that context, a good analytical framework and good modeling techniques must be able to handle the occurrence of singular behaviors whenever they are compatible with the mechanics or the physics of the problems at hand. In some optimization problems the natural intuitive notion of a geometric domain undergoes mutations into relaxed entities such as microstructures. So the objects under consideration need not be smooth open domains, or even sets, as long as they still make sense mathematically