Finite Order Approximations of Scattering Diagrams

Monday, June 1, 2015 - 2:30pm - 3:30pm
Lind 400
Greg Muller (University of Michigan)
Every seed in a cluster algebra determines a 'scattering diagram', a (possibly infinite) fan with additional information. By counting certain 'broken lines' in this diagram, Gross, Hacking, Keel, and Kontsevich define a family of formal power series called 'theta functions'. If these series all converge, they define a basis for an algebra which extends the cluster monomials, producing the kind of strongly positive basis that cluster algebras were introduced to explore.

A obstacle to this emerging theory is the (typically infinite) complexity of the scattering diagram. However, it is a limit of 'finite order approximations'; finite scattering diagrams with nilpotent parameters. Explicitly computing a scattering diagram to any finite order is a straightforward (but potentially lengthy) computation suitable for computer implementation. This has several potential applications:

1) A finite order approximation of a scattering diagram of a cluster algebra may be thought of as a 'coarsening' of the g-vector fan. This allows us to approximate the combinatorics of the clusters 'globally', in contrast with mutation-based techniques, which explore the combinatorics 'locally'. In particular, we can explore the possibility of 'missing' components of the exchange graph.

2) Finite order approximations of scattering diagrams allow finite order approximations of the theta functions. There are many open questions about the general behavior of theta functions; existing results largely confirm they behave well in good situations. There is essentially no data for poorly behaved cluster algebras; a problem which could be remedied by computational implementation.
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