Kontsevich conjectured that the wrapped Fukaya category of any finite-type (Wein)stein manifold is Morita equivalent to a dg algebra of finite type, that is, the path algebra of a finite graded quiver with differential. I will outline a proof in three steps: (1) a local-to-global gluing description of Fukaya categories via Ganatra–Pardon–Shende, (2) a local model for Weinstein manifolds using arboreal singularities, whose Fukaya categories are finite-type by work of Nadler, and (3) a cofibration category structure on the category of dg categories, developed with Sangjin Lee and myself, which ensures that gluing preserves finite-typeness. I will also explain the necessary background and definitions along the way.

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Kyoto Symplectic Seminar
https://sites.google.com/view/kyoto-symplectic/