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Cohomological Donaldson-Thomas theory

This review gives an introduction to cohomological Donaldson-Thomas theory: the study of a cohomology theory on moduli spaces of sheaves on Calabi-Yau threefolds, and of complexes in 3-Calabi-Yau categories, categorifying their numerical DT invariant. Local and global aspects of the theory are both covered, including representations of quivers with potential. We will discuss the construction of the DT sheaf, a nontrivial topological coefficient system on such a moduli space, along with some cohomology computations. The Cohomological Hall Algebra, an algebra structure on cohomological DT spaces, will also be introduced. The review closes with some recent appearances, and extensions, of the cohomological DT story in the theory of knot invariants, of cluster algebras, and elsewhere.