Research Project:
We propose to build the theory of totally noncommutative cluster structures and their symmetries. We also plan to find "purely" noncommutative analogs of the main results obtained in the commutative/quantum case and to relate them to the Donaldson-Thomas theory developed by Kontsevich and Soibelman, and other versions of noncommutative clusters recently discussed by Kontsevich and Goncharov. In particular, we propose to define noncommutative clusters, their mutations, the mutation-invariant automorphism groups (a generalization of mapping class groups for surfaces), and to construct important classes of noncommutative cluster structures such as principal noncommutative seeds and seeds for noncommutative double Bruhat cells introduced in our earlier paper, and also noncommutative Q-systems relevant to our recent joint paper where we introduced purely noncommutative Catalan numbers. As an application, we expect to obtain noncommutative analogs of (upper) cluster algebras, generalized mapping class groups, and, ultimately, the Noncommutative Laurent Phenomenon.

Research Team: Arkady Berenstein (U of Oregon), Vladimir Retakh (Rutgers U)

Original Dates: July 1 - August 31, 2020, postponed to 2022 due to COVID-19