[math-fun] Who's Saari now

A delated reaction from an expert in cosmological N-body simulations: "I think [Saari's] is a mathematically fascinating configuration and a nice example that assumptions on smooth/spherical distributions of points can fail catastrophically sometimes. This reminds me of the classic "exploding galaxy" problem, that Salmon & Warren (1994) constructed to demonstrate how a gravitational N-body code based on a tree-code algorithm can lead to unphysical results." "Still, much like we have been using tree codes for simulations without obtaining catastrophic results for decades, I think that the Saari's configuration is very special and not what is realized in galaxies. To support this, we can use observations of disk galaxies at different inclination angles which show the presence of a fairly smooth distribution of stars, typically with a profile exponentially declining with increasing radius. Finally, from the mathematical point of view, it would be interesting to evaluate whether Saari's configuration is stable against perturbations." ----------------- https://www.siam.org/pdf/news/2094.pdf Rethinking “Star Soup” By Dana Mackenzie One of the greatest mysteries in astrophysics these days is dark matter—-an invisible form of matter that has been detected only through its gravitational effects. According to the latest estimates from the European Space Agency’s Planck space telescope, dark matter accounts for 26.8% of the matter­energy in the universe. That is 5.5 times the amount of ordinary, or baryonic matter. (Both are dwarfed by “dark energy,” but that’s a subject for another article.) To date, nobody has ever actually seen a “lump” of dark matter. All the evidence for its existence depends on mathematical calculations—-something that ought to make mathematicians happy. But Don Saari, a longtime special- ist in celestial mechanics at the University of California at Irvine, is far from convinced. At a standing-room-only lecture at this year’s Joint Mathematics Meetings, Saari called into question a classical—-in fact the earliest—-method for estimating the amount of dark matter, which uses galactic rotation curves. “It’s a mathematical computation, and that computation is not correct,” Saari says. http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.407 Mathematics and the “Dark Matter” Puzzle Donald G. Saari Abstract. Surprisingly, aspects of the compelling mystery of “dark matter”—-the mysterious, undiscovered material that supposedly consumes most of a spiral galaxy’s mass—-are math- ematical issues rather than astronomical ones. When analyzed from the perspective of math- ematics, doubt is cast on standard predictions about the existence of huge amounts of this material.