Differential Geometry Seminar: A tale of two polytopes (related to geodesic flows on spheres)

Speaker: Holger Dullin (University of Sydney)

Title: A tale of two polytopes (related to geodesic flows on spheres)

Abstract: Separation of variables for the geodesic flows on spheres leads to a large
family of integrable systems whose integrals are defined through the separation constants. 
Reduction by the periodic flow of the Hamiltonian leads to integrable systems on Grassmanians. 
Specifically for the geodesic flown on the $S^3$ the reduced system defines a family of 
integrable systems on $S^2\times S^2$. We show that the image of these systems under
a continuous momentum map defined through the action variables has a triangle as its image. 
Each member of the family can be identified with a point inside a Stasheff polytope. 
Corners of the polytope correspond to toric systems (possibly with degenerations), 
edges correspond semi-toric systems (in various meanings of the word), 
and the face corresponds to ``generic'' integrable systems.