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Exact results in supersymmetric gauge theories in various dimensions

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One of the central challenges in theoretical physics is to develop non-perturbative methods to describe quantitatively the dynamics of strongly coupled quantum fields. Much progress in this direction has been made for theories with a higher degree of symmetry, such as conformal symmetry or supersymmetry. In recent years the method of localisation has allowed to obtain a great deal of exact results for supersymmetric gauge theories in various dimensions which has led to the discovery of new surprising correspondences such as the celebrated Alday-Gaiotto-Tachikawa correspondence. I will review some recent results which indicate that partition functions of supersymmetric theories formulated on compact manifolds can be expressed in terms of a small set of fundamental building blocks.