Polylogarithms, Cluster Algebras, and Scattering Amplitudes

September 11, 2023 - September 15, 2023


Dani Kaufman
Copenhagen University
Herbert Gangl
Durham University
Zack Greenberg
Heidelberg University
Christian Zickert
University of Maryland

The classical polylogarithms were studied in the 18th and 19th century by many prominent mathematicians including Abel, Euler, Kummer and Lobachevsky. They were mainly interested in special values and functional relations. In the 20th century deep relations to algebraic K-theory, characteristic classes and motivic cohomology were discovered for the dilogarithm, and conjectural generalizations were formulated. In the 21st century it was discovered that formulas for scattering amplitudes often involve polylogarithms evaluated at cluster coordinates for a Grassmannian. This brings the theory of cluster algebras to the study of polylogarithms. There are many exciting recent developments including the proof of Zagier's conjecture (expressing the regulator in terms of classical polylogarithms) in weight 4 by Goncharov and Rudenko (2018) following a depth reduction formula by Gangl (2016), the general depth reduction (to half the weight) by Rudenko (2020, formerly a conjecture of Goncharov), the precise formulation of cluster polylogarithms and depth reduction in weight 6 by Matveiakin and Rudenko (2022), a cluster formulation of the second motivic Chern class by Goncharov and Kislinskyi (2021), and iterated integral expressions for Grassmannian and Aomoto polylogarithms by Charlton, Gangl and Radchenko (2019).

This workshop seeks to shed light on the mysterious connection between multiple polylogarithms, cluster algebras, and scattering amplitudes by bringing together experts in these seemingly unrelated fields. Other topics to be discussed include Zagier's Polylogarithm Conjecture, depth reduction, Hodge correlators, and quantum polylogarithms.