The Donaldson–Thomas invariants of a Calabi–Yau threefold X are certain virtual curve counts produced using the moduli stack of stable sheaves on X. The MNOP conjecture, recently proven for most Calabi–Yau threefolds by Pandharipande and Pixton, predicts a tight relationship between Donaldson–Thomas and Gromov–Witten invariants. In this course, we will give an introduction to Donaldson–Thomas theory and discuss a refinement that produces invariants of motivic nature. To achieve this, we will study and apply the following tools: the local structure of quasi-smooth derived stacks with (-1)–shifted symplectic structure, the formalism of motivic vanishing cycles, and microlocal geometry.
Prerequisites: Algebraic geometry. Familiarity with homotopical algebra / infinity-categories, and mixed motives or ℓ-adic sheaves, is recommended.