Invitation to orbifold

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The notion of orbifold was first introduced by Satake under the name V-manifold as a generalization of manifold in the 1950s. Satake generalized several classic theorems in differential geometry (de Rham, Gauss-Bonnet) to the realm of orbifolds. Thurston coined the name orbifold in 70s during his lectures in Princeton, where he defined and showed the universal covering of orbifold exists.  Later on, using groupoid, Moerdijk gives a more subtle description on the cohomology and homotopy theory of orbifolds. In this presentation, I will give the basic notions of orbifolds and sketch Thurston's proof of existence theorem of universal covering.