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Description of  Diffeology (Mathematical Surveys and Monographs)

At the end of the last century, differential geometry was challenged by theoretical physics: new objects were displaced from the periphery of the classical theories to the center of attention of the geometers. These are the irrational tori, quotients of the 2-dimensional torus by irrational lines, with the problem of quasi-periodic potentials, or orbifolds with the problem of singular symplectic reduction, or spaces of connections on principal bundles in Yang-Mills field theory, also groups and subgroups of symplectomorphisms in symplectic geometry and in geometric quantization, or coadjoint orbits of groups of diffeomorphisms, the orbits of the famous Virasoro group for example. All these objects, belonging to the outskirts of the realm of differential geometry, claimed their place inside the theory, as full citizens. Diffeology gives them satisfaction in a unified framework, bringing simple answers to simple problems, by being the right balance between rigor and simplicity, and pushing off the boundary of classical geometry to include seamlessly these objects in the heart of its concerns.