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Thursday, April 1, 2021

**Abstract:** SubRiemannian Geometry studies triples consisting of a smooth manifold, a subbundle of the tangent bundle (a "tangent distribution"), and a metric along the distribution. Such a triple can be regarded as a degenerate limit of Riemannian manifolds, where the directions not in the distribution have been infinitely penalised. In such a setup, which is the framework in which Geometric Control Theory is phrased, a central question is to study the properties of the minimising curves. In many ways, this resembles the usual theory of geodesics in Riemannian Geometry, but various exotic behaviours appear. The underlying reason behind these behaviours is that, upon dualising, the subRiemannian metric becomes a degenerate Hamiltonian with fibrewise non-compact level sets. I will review the basic theory behind this setup, particularly the cotangent formulation, which goes back to work of Pontryagin. I aim to give a (very biased) overview of the area, emphasising various results about global topological properties of subRiemannian geodesics, as well as some intriguing open questions. If time allows, I will comment on recent work, joint with L. Dahinden, in which we study the subRiemannian billiard flow, which is the natural generalisation to manifolds with boundary.