Polylogarithms, cluster algebras, and hyperbolic geometry

This award supports research on the interplay between three different research areas: polylogarithms, cluster algebras, and hyperbolic geometry. Polylogarithms generalize the natural logarithm and have been studied since the 18th century. Cluster algebras, invented in the early 21st century, are purely combinatorial objects which are widely studied and broadly applicable. Hyperbolic geometry is a geometry with constant negative curvature, where Euclid's fifth postulate fails. Recent advances have revealed surprising links between these areas. For example, formulas for scattering amplitudes in high energy physics frequently involve polylogarithms evaluated at cluster algebra coordinates. Also, the volume of a certain hyperbolic polyhedron known as an orthoscheme, where successive faces form right angles, is given by a polylogarithm formula. The proposal will investigate key conjectures, find new examples of hyperbolic manifolds, and compute invariants using cluster coordinates. The PI will involve both graduate and undergraduate students in this project and continue his outreach to local schools. The proposal will explore the relationship between polylogarithms and cluster algebras focusing on several key conjectures in the field. These include the Matveiakin-Rudenko conjecture, that all polylogarithm relations arise from the cluster polylogarithm relations of type A_n; Zagier's polylogarithm conjecture, that the zeta function of a number field at integers is expressed by polylogarithms; and Goncharov's depth conjecture, that a polylogarithm is a classical polylogarithm if an only if its truncated coproduct vanishes. The proposal will explore special cases of these conjectures using Matveiakin and Rudenko's notion of cluster polylogarithms as well as new tools developed by the PI and his collaborators. In addition, the proposal will study Rudenko's polylogarithm formula for a hyperbolic orthoscheme, find new examples of hyperbolic manifolds that don't arise from Coxeter groups (and therefore have dihedral angles that are not a submultiple of pi), and generalize formulas for Cheeger-Chern-Simons invariants from dimension 3 to dimension 5. 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.