Journal of Symbolic Logic

Computability results used in differential geometry

Barbara F. Csima and Robert I. Soare

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

Abstract

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.

Article information

Source
J. Symbolic Logic Volume 71, Issue 4 (2006), 1394-1410.

Dates
First available: 20 November 2006

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1164060462

Digital Object Identifier
doi:10.2178/jsl/1164060462

Mathematical Reviews number (MathSciNet)
MR2275866

Zentralblatt MATH identifier
1109.03035

Citation

Csima, Barbara F.; Soare, Robert I. Computability results used in differential geometry. Journal of Symbolic Logic 71 (2006), no. 4, 1394--1410. doi:10.2178/jsl/1164060462. http://projecteuclid.org/euclid.jsl/1164060462.


Export citation

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.
  • --------, 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.