This blog will be paused again, this time until august probably, as I have no time to write posts nor moderate comments at the moment…
Archive for June, 2007
Further blog pause
June 17, 2007Secure billiards and Serre’s thesis
June 4, 2007A few days ago, Sergei Tabachnikov put a preprint on the arXiv called Birkhoff billiards are insecure. Now that’s a catchy title, so I had to at least browse the paper.
The thing is, it immediatly rang a bell, since the abstract says:
- We prove that every compact plane billiard, bounded by a smooth curve, is insecure: there exist pairs of points such that no finite set of points can block all billiard trajectories from to
while I had read last year a nice history book by Marcel Berger which provides an overview of mathematics in France in the past five centuries (I recommend it to french high-school students), and where it is written (my translation):
- already with his thesis in 1951, Serre became famous by using the Leray spectral sequence to show […] that [for any compact Riemannian manifold] any pair of points can be joined by an infinity of geodesics (Those lucky enough to have a JSTOR login can read Serre’s thesis here. Those who don’t can still view large excerpts on google books.)
Obviously there’s a striking similarity between the two. Tabashnikov defines a manifold to be secure if for any pair of points there exist a finite set of points such that all geodesics from to go through a point of .
So Serre’s result implies a statement similar to Tabashnikov’s abstract, but for compact riemannian manifolds: they are all insecure.
EDIT: JSTOR link corrected, added google books link.