Abstract

Many classical objects in differential geometry are described by soliton equations: nonlinear PDEs with infinitely many conserved quantities that are (in some sense) solvable. Since the 1980s, studies of curve evolutions that are invariant under a group of transformations have unveiled more connections between geometric flows and well-known integrable PDEs, such as the KdV, mKdV, sine-Gordon, and NLS equations.  More recent studies have addressed discrete analogues coming from geometric discretizations of surfaces and curves, and associated evolutions. I will discuss a few natural geometric flows for curves and polygons, highlighting the role of moving frames in integrability.  The main examples are the vortex filament flow in Euclidean geometry and its relation to the NLS equation, Pinkall’s flow in centroaffine geometry and the KdV equation, and discretizations of the Adler-Gel’fand Dikii flows for curves in projective space. This talk is based on joint work with Tom Ivey and Gloria Marí-Beffa.