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AG seminar by Karandeep Singh

AG seminar by Karandeep Singh

Stability problems and differential graded Lie algebras

Abstract: Stability problems appear in various forms throughout geometry and algebra. For example, given a vector field $X$ on a manifold that vanishes in a point, when do all nearby vector fields also vanish somewhere? As an example in algebra, we can consider the following question: Given a Lie algebra $\mathfrak g$, and a Lie subalgebra $\mathfrak h$, when do all deformations of the Lie algebra structure on $\mathfrak g$ admit a Lie subalgebra close to $\mathfrak h$? I will show that both questions are instances of a general question about differential graded Lie algebras, and under a finite-dimensionality condition which is satisfied in the situations above, I will give a sufficient condition for a positive answer to the general question. I will then discuss the application to fixed points of Lie algebra actions.