The Duflo isomorphism in Lie theory is an isomorphism between the invariant part of the symmetric algebra of a given Lie algebra, and the centre of its universal enveloping algebra. In 2003, Bar-Natan, Le and Thurston gave a topological proof for metrised Lie algebras. In this talk we give a topological proof in the general case, using a universal finite type invariant for certain knotted tubes in R^4, constructed recently by Bar-Natan and the speaker. If time allows, we'll discuss the broader connection to Lie theory: the (topological) Kashiwara-Vergne problem. (Joint work with Dror Bar-Natan.)