Mirror symmetry is a phenomenon predicted by string theory. It allows one to translate questions in symplectic geometry to questions in complex geometry, and vice versa. The homological mirror symmetry program interprets mirror symmetry within the unifying categorical framework of derived noncommutative geometry. After introducing these ideas, I will describe an approach to a theory of Kähler metrics in derived noncommutative geometry. We will see how this leads to (i) a non-Archimedean categorical analogue of the Donaldson-Uhlenbeck-Yau theorem, inspired by symplectic geometry, and (ii) the discovery of a refinement of the Harder-Narasimhan filtration which controls the asymptotic behavior of certain geometric flows. This talk is based on joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich.