Positive geometries are geometric objects at the interface between physics and mathematics, born in the study of scattering amplitudes. The usual definition of a positive geometry consists of a real semialgebraic set together with a complex meromorphic differential form—the canonical form—uniquely characterized by recursive properties on its boundaries. A prototypical example is the amplituhedron, a semialgebraic set in the Grassmannian whose canonical form computes the integrand of scattering amplitudes in planar N=4 super Yang–Mills theory.
In this talk, I introduce a dual notion of positive geometries, in which the central object is no longer a differential form but rather a probability distribution supported on a dual semialgebraic set. From this perspective, the canonical form effectively computes a volume associated with this distribution. This framework naturally explains—and geometrizes—the strong positivity properties of amplitudes and related observables that have been recently uncovered.