# 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
ÌÄ± Beffa (University of Wisconsin- Madison).

## Room Reservation Information

**Room Number:** 106 McAllister

**Date:** 11/05/2013

**Time:** 2:30pm - 3:30pm