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Friday, March 13, 2020

**Abstract:** Poincare duality of manifolds is a classical theorem which can be phrased in terms of the homology and cohomology groups of manifolds. However, when we look at singular spaces, this fails to hold for the usual homology and cohomology groups. In the setting of a certain class of singular spaces know as topological pseudomanifolds, which include orbifolds, algebraic varieties, moduli spaces and many other natural objects, one can extend these groups in order to recover some form of Poincare duality. I'll present how this was achieved by Goresky and MacPherson with their Intersection homology, and by Cheeger using L^2 cohomology and explain how they are related to each other, in similar spirit to the equivalence in the smooth setting. I'll only assume a basic knowledge of homology and cohomology.