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Annalisa Calini (College of Charleston, USA)

Geometric flows in centro-affine space
Tuesday, September 22, 2020, 15:00 GMT
Abstract: Many classical objects in differential geometry are described by integrable systems: nonlinear PDE with infinitely many conserved quantities that are (in some sense) solvable. Beginning in the 1980s, studies of curve evolutions that are invariant under the action of a geometric group of transformations have unveiled more connections between geometric curve flows and well-known integrable PDE (among them are the KdV, mKdV, sine-Gordon, and NLS equations). More recently, efforts have been directed towards understanding geometric discretizations of surfaces and curves and associated evolutions.

This talk will describe some natural geometric flows for curves and polygons in centro-affine geometry, and their relations with the KdV and Boussinesq equations and discretizations of Adler-Gel’fand-Dikii (AGD) flows for curves in projective space. The AGD discretizations (previously introduced in work by Marí-Beffa and Wang) will be discussed in more detail, as their lifts to the moduli space of centro-affine arc length parametrized polygons promptly reveal their bihamiltonian structures in terms of a pair of simple pre-symplectic forms.

Some references:

U. Pinkall, Hamiltonian flows on the space of star-shaped curves, Result. Math. 27 (1995), 328–332.

A. Calini, T. Ivey, and G. Marí-Beffa, Remarks on KdV-type Flows on Star-Shaped Curves, Physica D Vol. 238, no. 8 (2009), 788–797

G. Marí-Beffa and J.P. Wang, Hamiltonian structures and integrable evolutions of twisted gons in RPn, Nonlinearity 26 (2013) 2515-2551

A. Calini, T. Ivey, and G. Marí-Beffa, An integrable flow for starlike curves in centroaffine space, SIGMA 9, (2013), 022, 21 pp.

A. Calini and G. Marí-Beffa, Integrable evolutions of twisted polygons in centroaffine Rm, IMRN, rnaa161 (2020)