Integrable Curve Flows in Centroaffine Geometry

GAP Seminar

Meeting Details

For more information about this meeting, contact Mathieu Stiénon, Nigel Higson, Ping Xu.

Speaker: Annalisa Calini, College of Charleston, SC

Abstract: I will discuss integrable evolution equations for closed curves in centroaffine geometry. The planar setting is motivated by Pinkall's flow, a Hamiltonian evolution equation for closed star-shaped planar curves closely related to the KdV equation, and whose projectivization is the Schwarzian KdV equation. I will describe the relation between invariant curve evolutions in projective and centro-affine geometry and the associated geometric Hamiltonian structures, and construct examples of closed solutions of Pinkall's flow corresponding to periodic finite-gap KdV potentials. In the three-dimensional setting, I will describe the construction of integrable hierarchies of curve evolutions through a natural pre-symplectic structure on the space of closed unparametrized starlike curves. The induced evolution equations for the differential invariants are closely connected with the Boussinesq hierarchy, and the restricted hierarchy of flows on curves that project to conics in $\mathbb{RP}^2$ induces the Kaup-Kuperschmidt hierarchy at the curvature level. This is joint work with Tom Ivey (College of Charleston), and Gloria Mar ́ı Beffa (University of Wisconsin- Madison).

Room Reservation Information

Room Number: 106 McAllister

Date: 11/05/2013

Time: 2:30pm - 3:30pm