 ArXiv Preprints and unpublished

All ArXiv

Failed preprints:
• 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 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 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 in a stressful time with severe time constraints and unmovable deadlines.
• A deterministic displacement theorem [PDF] This was originally an attempt to prove the open problem that the Hamiltonian n-body problem has almost everywhere solutions (also non-collision singularities have measure zero). Let me add an erratum: "The statement in this displacement theorem needs to be modified. One either has to look at a Vlasov-BBGKY hierarchy or then make some additional randomness assumption. Without that, correlations will 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 preprint had 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 is an other realm of dynamical system, namely 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 would be much more technical. 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. See this list of open problems. The problem of singular potential is no problem 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). But there is still 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).

Oliver Knill, Department of Mathematics, Harvard University, One Oxford Street, Cambridge, MA 02138, USA. SciCenter 432 Tel: (617) 495 5549, Email: knill@math.harvard.edu Quantum calculus blog, Twitter, Youtube, Vimeo, Linkedin, Scholar Harvard, Academia, Harvard Academia, Google Scholar, ResearchGate, Slashdot, Ello, Webcam, Fall 2018 office hours: TBA Mon-Fri 11:30-12:30 AM and by appointment.