Prerequisites: abstract algebra, complex analysis, and some programming experience.
The theory of cluster algebras is a growing field of research with applications in many areas of mathe- matics including combinatorics, representation theory, and Teichmuller theory. The aim of the project is to use the theory of cluster algebras to explore the structural properties and func- tional relations of polylogarithms. The polylogarithms are generalizations of the logarithm and appear in hyperbolic geometry, algebraic K-theory, and also in formulas for scattering amplitudes in high energy physics. Their functional relations are currently mysterious, but growing evidence suggests that cluster algebras could hold the key to further progress. Students will learn the basics about cluster algebras including the cluster structure of the Grassmannian, as well as basic theory about polylogarithms. The students will explore the cluster polylogarithms appearing in high energy physics, and will write computer code to discover new cluster polylogarithms and new cluster relations.