Extremal length and duality

Abstract

Classical extremal length (or conformal modulus) is a conformal invariant involving families of paths on the Riemann sphere. In ``Extremal length and functional completion'', Fuglede initiated an abstract theory of extremal length which has since been widely applied. Concentrating on duality properties and applications to quasiconformal analysis, we demonstrate the flexibility of the theory and present recent advances in three different settings: Extremal length and uniformization of metric surfaces, Extremal length of families of surfaces and quasiconformal maps between $n$-dimensional spaces, and Schramm's transboundary extremal length and conformal maps between multiply connected plane domains.