| Abstract: |
| Of central importance in the n-body problem is the fact that isolated binary collisions can be regularised; a singular
change of space and time variables allows trajectories to pass analytically through binary collisions unscathed. This so
called Levi-Civita regularisation provides a flow smooth with respect to initial conditions. Curiously, when two binary
collisions occur simultaneously, we are not so fortunate. In 1999, Martinez and Sim\`{o} gave strong evidence to conjecture
the regularised flow, in a neighbourhood of the simultaneous binary collision, is at best C^{8/3}. Remarkably, the conjecture
has been shown for some sub-problems of the 4-body problem, including the collinear and trapezoidal problems.
In this talk we will provide a proof for the conjecture in the planar 4-body problem. Some notable components of the proof are
the use of a normal form procedure, a type of projective blow-up which produces a collision manifold foliated
by invariant \mathbb{RP}^3, and the study of transitions near manifolds of normally hyperbolic fixed points. Through the
proof, a link is established between the inability to construct a set of integrals local to simultaneous binary collisions and
the curious loss of differentiability. |
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