Positive Geometry

The aim of this project is to develop "positive geometry". Positive geometry was first conceived of in the study of fundamental questions in particle physics: the calculation of scattering amplitudes that determine how elementary particles, such as electrons and photons, interact. Positive geometries are shapes (for example, higher dimensional versions of cubes and pyramids) whose structure reflects the behavior of particle interaction. In this project, the PI will develop the mathematical foundations of positive geometry which will in turn be applied to physical questions. The project will involve both undergraduate and graduate students. Positive geometries are semi-algebraic spaces equipped with a differential form, the "canonical form", whose polar structure reflects the facial structure of the geometry. Examples of positive geometries include convex polytopes, positive parts of toric varieties, totally nonnegative flag varieties, and conjecturally Grassmann polytopes and amplituhedra. This project aims to study the combinatorics, topology, and geometry of positive geometries in analogy with the theory of convex polytopes. The project will find new positive geometries and new formulae for canonical forms, and apply this to the theory of scattering amplitudes in physics. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.