Archive for June, 2007

Further blog pause

June 17, 2007

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…

Secure billiards and Serre’s thesis

June 4, 2007

A 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 $A,B$ such that no finite set of points can block all billiard trajectories from $A$ to $B$

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 $M$ to be secure if for any pair of points $A, B$ there exist a finite set of points $S\subset M-\{A,B\}$ such that all geodesics from $A$ to $B$ go through a point of $S$.

So Serre’s result implies a statement similar to Tabashnikov’s abstract, but for compact riemannian manifolds: they are all insecure.