MaPSS: Maths Postgraduate Seminar Series: Nathan Duignan -- Simultaneous Binary Collisions and the Mysterious 8/3

Hello all,

The next MaPSS talk of this semester will be at 17:00 on Monday 18th March in Carslaw
535. It’s a great opportunity to meet fellow postgrads, listen to an interesting talk,
and of course get some free pizza!

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Speaker: Nathan Duignan

Title: Simultaneous Binary Collisions and the Mysterious 8/3

Abstract: Of central importance in the N-body problem is the fact that isolated binary 
collisions can be regularized :  that a singular change of space and time variables 
(first written down by Levi-Civita) allows trajectories to pass analytically through 
binary collisions unscathed. The resulting flow is smooth with respect to initial 
conditions. Curiously, we are not so lucky with simultaneous binary collisions. In a 
landmark paper, Martinez and Simo gave strong evidence that the best one can hope 
is 8/3 differentiability of the flow in a neighborhood of simultaneous binary collisions 
in the 4 body problem. In this talk we follow Easton by linking regularizability to the 
behaviour of the flow in Conley isolating blocks around the collisions. We show 
the 8/3 is produced from the first resonant monomial with nonzero coefficient near 
a degenerate singularity formed when the two binaries are separately Levi-Civita 
regularized. To show this, we blow-up the singularity and study the flow near the 
resulting two, 3:1 resonant, normally hyperbolic manifolds connected by heteroclinics. 
A lengthy normal form computation confirms the conjecture.

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See you there!

Details can also be found on the school’s new Postgraduate Society website: 
http://www.maths.usyd.edu.au/u/MaPS/mapss.html 
 
Cheers,
Eric