The noncommutative geometry of difference equations

The existence of Lax pairs associated to the Painlev\'e equations gives rise to a natural interpretation of their spaces of initial conditions as moduli spaces of differential equations. This suggests that one should develop the theory of such moduli spaces more generally, in particular for difference equations of various kinds (including elliptic) as well as for differential equations. I'll describe how to translate those moduli problems into natural moduli problems arising in noncommutative geometry, how (discrete) isomonodromy deformations arise naturally in that setting, and some of the consequences for the elliptic Painlev\'e equation and generalizations.Non UBCUnreviewedAuthor affiliation: California Institute of TechnologyResearche