New Life of D-branes in Math
One of the most wonderful gifts from string theory to pure mathematics comes from Mike Douglas' ideas on the decay of D-branes and walls of marginal stability. Tom Bridgeland formalized structures discovered by Mike as stability conditions in abstract triangulated categories. This notion became central in modern homological algebra and is also pivotal in the theory of Donaldson-Thomas invariants. I'll review several (hypothetical) extensions of the original picture, involving Fukaya categories with coefficients, non-archimedean B-model, etc.