Real Analysis, Quantitative Topology, and Geometric Complexity - S. Semmes

Real Analysis, Quantitative Topology, and Geometric Complexity - S. Semmes

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: arXiv:math.MG/0010071 v1 7 Oct 2000 Real Analysis, Quantitative Topology, and Geometric Complexity Stephen Semmes This survey originated with the John J. Gergen Memorial Lectures at Duke University in January, 1998. The author would like to thank the Math- ematics Department at Duke University for the opportunity to give these lectures. See [Gro1, Gro2, Gro3, Sem12] for related topics, in somewhat different directions. Contents 1 Mappings and distortion 3 2 The mathematics of good behavior much of the time, and the BMO frame of mind 10 3 Finite polyhedra and combinatorial parameterization prob- lems 17 4 Quantitative topology, and calculus on singular spaces 26 5 Uniform rectifiability 36 5.1 Smoothness of Lipschitz and bilipschitz mappings . . . . . . . 42 5.2 Smoothness and uniform rectifiability . . . . . . . . . . . . . . 47 5.3 A class of variational problems . . . . . . . . . . . . . . . . . . 51 Appendices A Fourier transform calculations 54 The author was partially supported by the National Science Foundation. 1 B Mappings with branching 56 C More on existence and behavior of homeomorphisms 59 C.1 Wildness and tameness phenomena . . . . . . . . . . . . . . . 59 C.2 Contractable open sets . . . . . . . . . . . . . . . . . . . . . . 63 C.2.1 Some positive results . . . . . . . . . . . . . . . . . . . 67 C.2.2 Ends of manifolds . . . . . . . . . . . . . . . . . . . . . 72 C.3 Interlude: looking at infinity, or looking near a point . . . . . 72 C.4 Decomposition spaces, 1 . . . . . . . . . . . . . . . . . . . . . 75 C.4.1 Cellularity, and the cellularity criterion . . . . . . . . . 81 C.5 Manifold factors . . . . . . . . . . . . . . . . . . . . . . . . . . 84 C.6 Decomposition spaces, 2 . . . . . . . . . . . . . . . . . . . . . 86 C.7 Geometric structures for decomposition spaces . . . . . . . . . 89 C.7.1 A basic class of constructions . . . . . . . . . . . . . . 89 C.7.2 Comparisons between geometric and topological prop- erties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 C.7.3 Quotient spaces can be topologically standard, but ge- ometrically tricky . . . . . . . . . . . . . . . . . . . . . 96 C.7.4 Examples that are even simpler topologically, but still nontrivial geometrically . . . . . . . . . . . . . . . . . 105 C.8 Geometric and analytic results about the existence of good coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 C.8.1 Special coordinates that one might consider in other dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 113 C.9 Nonlinear similarity: Another class of examples . . . . . . . . 118 D Doing pretty well with spaces which may not have nice co- ordinates 118 E Some simple facts related to homology 125 References 137 2 1 Mappings and distortion A very basic mechanism for controlling geometric complexity is to limit the way that distances can be distorted by a mapping....
View Full Document

This note was uploaded on 04/07/2008 for the course MATH dont know taught by Professor Dontknow during the Spring '08 term at Case Western.

Page1 / 161

Real Analysis, Quantitative Topology, and Geometric Complexity - S. Semmes

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online