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DOI: 10.1016/s0304-0208(08)70181-6
  • Mathematics


Publisher Summary This chapter introduces the book on a wide range of concepts, ideas, results, and techniques that evolve around symmetry-groups, representations, and Laplacians. In Mathematics, symmetries play several important functions. They allow reducing the number of variables, and often rendering problem soluble. On the other hand, they allow analyzing and synthesizing complex objects in terms of simple (elemental) blocks/constituents. The chapter focuses on the diversity and versatility of the subject, but at the same time to unravel its common roots. The chapterprovides an overview of geometrical objects and structures {X}, discrete or continuous that possesses sufficiently large symmetry-group G, like regular graphs, lattices, symmetric Riemannian manifolds, and other. To explain and motivate the problems and the methods, based on symmetries, the chapter provides few simple prototypical examples.

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