Zack Greenberg

Contact

Email: greenberg@mis.mpg.de

Research Interests

RTG Lecture Series: Cluster Algebras For Geometers

My former lecture series as part of the RTG days at Karlsruhe Institute of Technology and Heidelberg University. My lecture notes can be found here: Cluster algebras for Geometers.

Papers

• Noncommutative Polygonal Cluster Algebras with D. Kaufman, M. Niemeyer and A. Wienhard
• Multiplicative Inequalities In Cluster Algebras Of Finite Type with M. Gekhtman and D. Soskin
• SL2-Like Properties of Matrices Over Noncommutative Rings and Generalizations of Markov Numbers with D. Kaufman and A. Wienhard
• Restrictions on Anosov subgroups of Sp(2n,R) with S. Dey and M. Riestenberg
• Hopf algebras of multiple polylogarithms, and holomorphic 1-forms with D. Kaufman, H. Li, C. Zickert
• The Lie coalgebra of multiple polylogarithms with D. Kaufman, H. Li, C. Zickert
• Cluster Modular Groups of Affine and Doubly Extended Cluster Algebras with D. Kaufman.

  The following link contains the computational reciepts for the proof of the A-coordinate structure of doubly extended cluster algebras

• Thesis: Cluster Algebras and Polylogarithm Relations

Cluster Algebra Explorer

The cluster algebra explorer is a tool I developed to visualize the structure of cluster algebras. The tool allows the user to highlight codimension 1 subalgebras and orbits of A and X coordinates under the cluster modular group.

Cluster Algebra Code

The ocaml library for cluster algebra computations can be found below. It is designed to generate and search cluster complexes and compute cluster modular groups. All the pictures above were generated using this and graphviz/gephi. There is specific support for grassmanian cluster combinatorics, but can work for arbitrary cluster algebras.

Personal Course Notes

My personal notes from class. Hopefully everything is accurate but no guarentees.