The talk concerns with a naive model of bicycle motion: a bicycle is a segment of fixed length that can move so that the velocity of the rear end is always aligned with the segment.

1) The rear wheel track and a choice of the direction of motion uniquely determine the front wheel track; changing the direction to the opposite, yields another front track. These two front tracks are said to be in a bicycle correspondence, which is an integrable discrete time dynamical system on the space of curves, closely related with with another completely integrable dynamical system, the filament (a.k.a binormal, smoke ring, local induction) equation. 

2) Given the rear and front tracks of a bicycle, can one tell which way the bicycle went? Usually, one can, but sometimes one cannot. The description of these ambiguous tire tracks is a problem which is only partially understood, and the known examples are solitons of the planar version of the filament equation. 

3) Bicycling geodesics are bicycle paths whose front track's length is critical among all bicycle paths connecting two given placements of the line segment. In the plane, these geodesic front tracks are elastica, and in space they are  Kirchhoff rods. The bicycling sub-Riemannian geodesic flow is complete integrable.

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