Positive Geometries and Wonderful Compactifications

A positive geometry consists of a real projective variety and a semialgebraic subset (its “positive part”), together with a canonical rational form which satisfies a recursive definition when restricted to the boundary of the semialgebraic set. Positive geometries have been objects of interest in physics, and have recently started being explored mathematically. In my talk, I will focus on hyperplane arrangements in projective space. Regions in a hyperplane arrangement complement are polytopes, which are known to be positive geometries. I will discuss when such a region remains a positive geometry after taking the wonderful compactification of the arrangement. This talk is based on work in progress with S. Brauner, C. Eur, and L. Pratt.