Solitons and many-body systems in algebraic geometry 
-- David Ben-Zvi, February 28, 2003 

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A puzzling discovery of the theory of integrable systems 
(due to Airault-McKean-Moser and Krichever) is that the
motion of the poles of meromorphic solutions to soliton equations (PDE
such as the Korteweg-deVries equation) is often governed by
integrable many-body systems (ODE such as the Calogero-Moser system).
I will present an explanation of this phenomenon (joint work with
T. Nevins) using (noncommutative) algebraic geometry.  We study the
space of "configurations of points on the quantum plane" and other
spaces of noncommutative vector bundles as a natural bridge between
solitons and particles. Namely, the soliton equations are realized as
flows on these configurations, and a geometric Fourier transform
converts the flows into the linear flows along tori (Jacobians of
spectral curves) which give the "integration" of the many-body
system.