Hochschild cohomology of dg categories, derived centers and spectral sequences

The Hochschild cohomology of a differential graded algebra or more generally of a differential graded category admits a natural map to the graded center of its derived category: the characteristic homomorphism. We interpret it as an edge homomorphism in a spectral sequence. This gives a conceptual explanation of the possible failure of the characteristic homomorphism to be injective or surjective, answering a question by Bernstein and Drinfeld. To illustrate this, we will discuss several examples from algebra, geometry and topology, like coherent sheaves over algebraic curves, cohomology of free loop spaces and string topology. This is joint work with Markus Szymik (NTNU Trondheim).