This talk is based on joint work with Haiden, Katzarkov and Kontsevich.
I will describe our attempts to formalize and understand the
mathematical structures underlying the physical notion of stability for
D-branes in string theory. Our work builds on Bridgeland's notion of
stability conditions, and is inspired by ideas from symplectic geometry,
non-Archimedean geometry, geometric invariant theory and gauge theory.