Ian Stewart
In nonlinear dynamics, the most significant phenomena are those that remain invariant under appropriate changes of coordinates — diffeo(morphism)s. In equivariant dynamics, for example, appropriate means preserving the symmetry, leading to the usual concept of an equivariant diffeo. What about network dynamics? The most sensible definition would seem to be preserving admissible maps. So what are the diffeos with this property? There are at least three reasonable ways for a diffeo to act on a map: composition on the right, composition on the left, and a vector fieldchange in which the inverse of the derivative of the diffeo acts on the left and the diffeo also acts on the right. In joint work with Marty Golubitsky, we have classified diffeos that preserve admissibility in all three cases, for networks in which no two arrows are equivalent. (This is an important, though somewhat neglected, condition: it is very common in real-world networks). The answer is different in each case, although the three are closely related. It depends on three special subnetworks: the left core, right core, and their intersection (core). Being admissible for the appropriate core is almost the answer, but not in the vector field case. The proofs are easy for left and right actions, and require some abstract algebra (notably the Wedderburn-Malcev Theorem for associative algebras) in the vector field case. We describe the problem, define the cores, outline the proofs, and make a guess about one potential application.
