Oliver Knill Preprints

ArXiv Stuff

This page has not been updated much. Most of my work is unpublished. Fortunate not to face the "publish or perish" dilemma, I only submit something if there is a compelling reason to do so.

Failed projects:

The Problem of Positive Kolmogorov-Sinai entropy for the Standard map This was an attempt to prove that the Standard map T(x,y) = (2x-y+c sin(x),y) on T² has metric entropy bounded below by log(c/2). The idea was to push Herman's subharmonic estimates to a real analytic situation (but not complex analytic frame work) using multi-linear algebra. The approach is described a bit more in this math table talk of 2004[PDF]. It also lead to This paper from 2000 which I find one of my best papers I have been writing as it introduces a novel homogenisation approach to estimate fluctuations of subharmonic functions (which is an extremely classical area of mathematics). All these papers were written during a stressful time with severe time constraints and unmovable deadlines (postdoc times have fixed timelines and after that time is over, the game is over). It is tempting in such situations to make a hail Mary pass using a difficult unsolved problem.
A deterministic displacement theorem [PDF] This was work on an important open problem in Hamiltonian dynamics. The goal had been to show that the Hamiltonian n-body problem has almost everywhere solutions (also non-collision singularities have measure zero). This is a much too underappreciated problem in Celestial mechanics. The most natural approach to the problem is to analyze non-collsion singularities where particles escape to infinity in finite time. But the construction of examples of such initial conditions was insane. An toerh approach therefore is to try to take a continuum measure of masses and to see this measure evolve using an integro-partial differential equation, then show that this measure describes the average of solutions of the real problem. I had hoped that the simplest mean field model, the Vlasov system would do the job. Evolve a Poisson cloud of particles using the n-body dynamics and show that the density moves according to Vlasov. "The statement in the displacement theorem announcement is incorrect. One would either have to look at a Vlasov-BBGKY hierarchy or then make additional randomness assumption. Without that, correlations develop which would then have to be evolved using higher order correlations etc. (This was pointed out to me by H. Spohn sometime in 1998). The announceement been an attempt to solve the open problem that the Hamiltonian Newtonian n-body problem has a solutions for almost all initial conditions. Unfortunately, the BBGKY stuff is technically complicated. When introducing randomness, one enters an other class of dynamical systems which Boltzman type equations. The work (which was done in 1996-1997 while I was in Arizona and Texas) was an opportunity to learn about Poisson processes as well as Vlasov dynamics. Section 5.4 in the probability book profited from this research. Tackling a BBGKY expansion similar to a Taylor expansion turned out to too technical for me. The result proven in that electronic announcement is that the first derivative for correlations is zero. This is correct. The conclusion that the correlations remain zero for later time is false. What actually happens is that higher derivatives are no more zero in general. Analyze higher derivatives leads to higher order Vlasov equations, a version of the so called BBGKY hierarchy. But this is then no more a Poisson process. The picture of evolving Poisson processes is tempting, as it interprets a probability measure as an average of finite point processes. My quest had been to see the Vlasov equation as an integral equation which describes the mean of a probability space of n-body problems. The existence of solutions of the Vlasov equation then would lead to almost everywhere existence of solutions of the n-body problems. The almost everywhere existence of solutions to the n-body problem is a famous open problem and part of Simon's problems. See this list of open problems. The problem of singular potential in the Newtonian problem is not the difficulty as there are existence theorems for Vlasov dynamics with the -1/|x| potential. Vlasov appears still a promising approach for that existence theorem (analyzing non-collision singularities looks in comparison extremely hard; even establishing existence of particles moving to infinity in finite time was a tour de force by mathematicians like Saari or Xia). There is hope that some mean field theory can help to estimate averages of multi-particle n-body processes and establish global existence for almost all initial conditions (one of the celebrated Barry Simon's open problems in mathematical physics).