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GDMSeminar: Abstracts of the Talks (2020/12)

Alexander P. Veselov  (Loughborough, UK and Moscow, Russia)

Geometrization, integrability and knots

Tuesday, November 3, 2020, 15:00 GMT

Abstract:  After Arnold the classical integrability is usually understood in the Liouville sense as the existence of sufficiently many Poisson commuting integrals. About 20 years ago it was discovered that this does not exclude the chaotic behaviour of the system, which may even have positive topological entropy.

I will review the current situation with Liouville integrability in relation with Thurston’s geometrization programme, using as the main example the geodesic flows on the 3-folds with SL(2,R)-geometry. A particular case of such manifold SL(2,R)/SL(2,Z) is known (after Milnor and Quillen) to be topologically equivalent to the complement of the trefoil knot in 3-sphere. I will explain that the remarkable results of Ghys about modular and Lorenz knots can be naturally extended to the integrable region, where these knots are replaced by the cable knots of trefoil.
The talk is partly based on a joint work with Alexey Bolsinov and Yiru Ye and will not require any specific knowledge.