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October 5 Math Colloquium

Annalisa Calini, CofC

Integrable Evolutions of Twisted Polygons in Centro-Affine Space

Many classical objects in differential geometry are described by integrable systems, nonlinear partial differential equations (PDE) with many conserved quantities that are (in some sense) solvable. For example, surfaces of constant negative curvature in Euclidean space are described by the sine-Gordon equation. Beginning in the 1980s, studies of curve evolution equations 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. In the case of curves, these lead to completely integrable maps on discrete curves (as the now famous Pentagram Map) or to continuous time evolutions of polygons. In this talk, I will focus on the latter: I will introduce a system of ODE that describes a natural geometric flow for polygons in centro-affine geometry and I will discuss its integrable structure.

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