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Abstract: In this talk, we will discuss the problem of defining a suitable class of connections to perform Calculus of Variations with the Yang-Mills functional in dimensions higher than 4. In particular, I will present a generalization of the class of weak connections (or L^2-connections) for U(1)-bundles introduced by T. Rivière and collaborators, and illustrate some of its properties. This is joint work with R. Caniato.