Stability problems and differential graded Lie algebras

Stability problems appear in various forms throughout geometry and algebra. In differential geometry, given a vector field on a manifold that vanishes in a point, one can ask when nearby vector fields also vanish in a point. In algebra, given a Lie algebra and a subalgebra, one can ask when all deformations of the ambient Lie algebra also admit a Lie subalgebra of the same dimension. I will show that both questions are instances of a general question about differential graded Lie algebras. Under a finite-dimensionality assumption which is satisfied in the examples above, I will give a sufficient condition for a positive answer to the general question. I will then discuss some applications.