Computability Theory and Differential Geometry

Computability theory and differential geometry

Let M be a smooth, compact manifold of dimension n ≥ 5 and sectional curvature |K| ≤ 1. Let Met(M) = Riem(M)/Diff(M) be the space of Riemannian metrics on M modulo isometries. Nabutovsky and Weinberger studied the connected components of sublevel sets (and local minima) for certain functions on Met(M) such as the diameter. They showed that for every Turing machine Te, e ∈ ω, there is a sequence...

Computability results used in differential geometry

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 ...

Ergodic Theory, Group Theory and Differential Geometry.

6Hall, B. D., and S. Spiegelman, these PROCEEDINGS, 47,137(1961). 7Nygaard, A. P., and B. D. Hall, Biochem. Biophys. Res. Comm., 12, 98 (1963). 8 Adams, M. H., Bacteriophages (New York: Interscience Publishers, Inc., 1959), p. 446. 9 Kellenberger. G., M. L. Zichichi, and J. J. Weigle, these PROCEEDINGS, 47, 869 (1961). 10 Weigle, J., M. Meselson, and K. Paigen, J. Mot. Biol., 1, 379 (1960). 11 ...

Computability in Computational Geometry

We promote the concept of object directed computability in computational geometry in order to faithfully generalise the wellestablished theory of computability for real numbers and real functions. In object directed computability, a geometric object is computable if it is the effective limit of a sequence of finitary objects of the same type as the original object, thus allowing a quantitative ...

Computability Theory