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The noncommutative geometry of difference equations

Banff International Research Station for Mathematical Innovation and Discovery

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.

Mathematics; Ordinary differential equations; Dynamical systems and ergodic theory; Dynamical systems

video/mp4

Author affiliation: California Institute of Technology

2017-04-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/61092

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/