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The Donaldson-Uhlenbeck-Yau theorem relates the stability of a holomorphic bundle on a Kähler manifold to the existence of a good Hermitian metric on the bundle. There is an analogous conjectural picture in symplectic geometry, where objects in Fukaya categories play the role of holomorphic bundles, and holomorphic differentials play the role of Kähler forms. I will outline a category-theoretic framework for studying these phenomena, and discuss some applications of this framework. This is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.

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