Mathematics Colloquia and Seminars

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With collisions deleted, the configuration space for the planar N-body problem becomes a K(\pi, 1) where \pi is the braid group, and so has many free homotopy classes. Are all these classes realized? The answer may depend on the angular momentum and the mass ratios. We describe a recent result asserting that the answer is `yes' for N= 3 provided the masses are equal or near equal and the angular momentum is small but non-zero. We conjecture that the answer is `no' with the same N and mass ratios, but for zero angular momentum case and provide some circumstantial evidence for this `no'. The geometric importance of angular momentum zero is that this is the situation in which variational methods descend nicely to the quotient of the configuration space by the symmetry group. Time, energy, and audience demand permitting, I may discuss two other topics. (1) How, if we `cheat' by changing the Newtonian potential to a 1/r^2 potential, a complete metric with negative holomorphic curvature arises [the Jacobi-Maupertuis metric] whose geodesics are the solutions at energy 0. (2) In the case of very high positive energy how a novel billiard system arises in which the ``obstacles'' are linear subspaces of codimension greater than one, and how its billiard trajectories can be understood using a beautiful class of (non-smooth) Hadamard spaces introduced by Burago, Ferleger and Kononenko.