Source: J. Symbolic Logic Volume 71, Issue 4
(2006), 1394-1410.
Topologists Nabutovsky and Weinberger discovered how to embed
computably enumerable (c.e.) sets into the geometry of Riemannian
metrics modulo diffeomorphisms. They used the complexity of the
settling times of the c.e. sets to exhibit a much greater complexity
of the depth and density of local minima for the diameter function
than previously imagined. Their results depended on the existence of
certain sequences of c.e. sets, constructed at their request by Csima
and Soare, whose settling times had the necessary dominating
properties. Although these computability results had been announced
earlier, their proofs have been deferred until this paper.
Computably enumerable sets have long been used to prove
undecidability of mathematical problems such as the word
problem for groups and Hilbert’s Tenth Problem. However, this example
by Nabutovsky and Weinberger is perhaps the first example of the use
of c.e. sets to demonstrate specific mathematical or geometric
complexity of a mathematical structure such as the depth and
distribution of local minima.
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription.
Read more about accessing full-text
References
G. Barmpalias and E. M. Lewis, The ibT degrees of computably enumerable sets are not dense, Annals of Pure and Applied Logic, vol. 141 (2006), pp. 51--60.
--------, Randomness and the Lipschitz degrees of computability, to appear.
S. B. Cooper, Computablility theory, Chapman & Hall/CRC Mathematics, London, New York, 2004.
B. F. Csima, Applications of computability theory to prime models and differential geometry, Ph.D. thesis, The University of Chicago, 2003.
B. F. Csima and R. A. Shore, The settling-time reducibility ordering, to appear.
R. Downey, D. Hirschfeldt, and G. LaForte, Randomness and reducibility, Mathematical foundations of computer science, MFCS 2001 (J. Sgall, A. Pultr, and P. Kolman, editors), Lecture Notes in Computer Science, vol. 2136, Springer, Berlin, 2001, pp. 316--327.
--------, Randomness and reducibility, Journal of Computer and System Sciences, vol. 68 (2004), pp. 96--114.
A. Nabutovsky and S. Weinberger, Variational problems for Riemannian functionals and arithmetic groups, Publications Mathématiques, Institut des Hautes Études Scientifiques, vol. 92 (2000), pp. 5--62.
--------, The fractal nature of Riem/Diff I, Geometrica Dedicata, vol. 101 (2003), pp. 1--54.
R. I. Soare, Recursively enumerable sets and degrees: A study of computable functions and computably generated sets, Springer-Verlag, Heidelberg, 1987.
Mathematical Reviews (MathSciNet):
MR882921
--------, Computability theory and differential geometry, Bulletin of Symbolic Logic, vol. 10 (2004), pp. 457--486.
--------, Computability theory and applications, Springer-Verlag, Heidelberg,to appear.
S. Weinberger, Computers, rigidity, and moduli. The large-scale frcatal geometry of Riemannian moduli space, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 2005.
