Reading

Nathan Reading

"Mutation-linear maps"

Linear algebra can be formulated as the study of linear relations. What happens if we distinguish a subset R of the set of linear relations and reformulate linear algebra by ignoring all linear relations not in R? (For example, an independent set in this sense is a set of vectors that doesn't support a nontrivial linear relation contained in R.) One might guess that the answer to the question "What happens?" should be "Nothing interesting." But in fact, the question is motivated by very interesting examples. The motivating examples - the only examples I know of - come from the study of cluster algebras. In these examples, we are studying mutation-linear algebra: studying linear relations that are preserved by matrix mutation. In earlier work, we showed that finding a basis, in the mutation-linear sense, corresponds to finding universal coefficients for cluster algebras.

This talk will focus on the notion of mutation-linear maps. In many cases, mutation-linear maps are closely related to coarsenings of fans, and in some cases the fans in question are the Cambrian fans (normal fans of generalized associahedra). There is a direct connection to lattice homomorphisms between Cambrian lattices and a heuristic connection to ring homomorphisms between cluster algebras. I will try to convey the main (mutation-linear-)algebraic, geometric, and combinatorial ideas surrounding mutation-linear maps, without assuming any prior knowledge of cluster algebras, matrix mutation, or Cambrian lattices/fans.