ICTS

Elementary particle scattering is perhaps the most basic physical process in Nature. The data specifying the scattering process defines a "kinematic space", associated with the on-shell propagation of particles out to infinity. By contrast the usual approach to computing scattering amplitudes, involving path integrals and Feynman diagrams, invokes auxilliary structures beyond this kinematic space--local interactions in the interior of spacetime, and unitary evolution in Hilbert space. This description makes space-time locality and quantum-mechanical unitarity manifest, but hides the extraordinary simplicity and infinite hidden symmetries of the amplitudes,g that have been uncovered over the past thirty years. The past decade has seen the emergence of a new picture, where scattering amplitudes are seen as the answer to an entirely different sort of mathematical question involving "positive geometries" directly in the kinematic space, making surprising connections to total positivity, combinatorics and geometry of the grassmannian, and cluster algebras. The hidden symmetries of amplitudes are made manifest in this way, while locality and unitarity are seen as derivative notions, arising from the "factorizing" boundary structure of the positive geometries. This was first see in the story of "amplituhedra" and scattering amplitudes in planar N=4 SYM theory. In the past few years, a similar structure has been seen for non-superysmmetric "bi-adjoint" scalar theories with cubic interactions, in any number of dimensions. The positive geometries through to one-loop order are given by "cluster polytopes"--generalized associahedra for finite-type cluster algebras--with a simple description involving "dynamical evolution" in the kinematic space. Extending these ideas involves understanding cluster algebras associated with triangulations of general Riemann surfaces. These cluster algebras are infinite, reflecting the infinite action of mapping class group. One of the manifestations of this infinity is that the "g-vector fan" of the cluster algebra is not space-filling, making it impossible to define cluster polytopes, and obstructing the connection with positive geometries and scattering amplitudes. Remarkably, incorporating non-cluster variables, associated with closed loops in the Riemann surfaces, suggests a natural way of modding out by the mapping class group, canonically compactifying the cluster complex, and associating it with "clusterhedron" polytopes. Clusterhedra are conjectured to exist for all surfaces, providing the positive geometry in kinematic space for scattering amplitudes in the bi-adjoint scalar theory to all loop orders and all orders in the 1/N expansion. In this talk I will give an overview of this set of ideas, assuming no prior knowledge of scattering amplitudes or cluster algebras. My lectures at the ICTS workshop on scattering amplitudes later this month, will give a systematic, self-contained exposition of all the physical and mathematical ideas involved.