Abstract

The past decade has seen the emergence of surprising new connections between the real-world physics of elementary particle scattering processes, and the notion of “positive geometries”, linking combinatorics, algebra and geometry in interesting new ways, and involving a number of appearances of total positivity and cluster structure.  These ideas provide, in a number of examples, a different starting point for conceptualizing this physics, where the fundamental principles of spacetime and quantum mechanics are not taken as primary, but instead emerge from a more primitive mathematical rubric. In this talk I will illustrate these ideas in their simplest setting, showing how the most basic particle scattering amplitudes are determined by “surfacehedra”, certain generalizations of associahedra and finite-type cluster polytopes,  associated with any surface. I will also describe how the combinatorial relationships captured by the facets of these polytopes remarkably admit a “curvy” realization in terms of “binary geometries”, which provides a global description of the boundary structure of teichmuller space, and has the physical interpretation of generalizing from particle to string scattering”.