Speaker: Amanda Burcroff (Harvard)

Title: Scattering Diagram Combinatorics and Positivity for Generalized Cluster Algebras

Abstract:

Cluster algebras are celebrated for their intriguing positivity properties. Two distinct proofs of this positivity have emerged: one through the combinatorics of Dyck paths, and another via scattering diagrams, which originate from mirror symmetry and were previously not combinatorially understood.  Combining these approaches, we find a directly computable, manifestly positive, and elementary but highly nontrivial formula describing rank 2 scattering diagrams. Using this, we prove the Laurent positivity of generalized cluster algebras of all ranks, resolving a conjecture of Chekhov and Shapiro from 2014. This is joint work with Kyungyong Lee and Lang Mou.