Computers, Rigidity, and Moduli: The Large-scale Fractal Geometry of Riemannian Moduli Space

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Princeton University Press, 2005 - Mathematics - 174 pages
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This book is the first to present a new area of mathematical research that combines topology, geometry, and logic. Shmuel Weinberger seeks to explain and illustrate the implications of the general principle, first emphasized by Alex Nabutovsky, that logical complexity engenders geometric complexity. He provides applications to the problem of closed geodesics, the theory of submanifolds, and the structure of the moduli space of isometry classes of Riemannian metrics with curvature bounds on a given manifold. Ultimately, geometric complexity of a moduli space forces functions defined on that space to have many critical points, and new results about the existence of extrema or equilibria follow.


The main sort of algorithmic problem that arises is recognition: is the presented object equivalent to some standard one? If it is difficult to determine whether the problem is solvable, then the original object has doppelgängers--that is, other objects that are extremely difficult to distinguish from it.


Many new questions emerge about the algorithmic nature of known geometric theorems, about "dichotomy problems," and about the metric entropy of moduli space. Weinberger studies them using tools from group theory, computability, differential geometry, and topology, all of which he explains before use. Since several examples are worked out, the overarching principles are set in a clear relief that goes beyond the details of any one problem.


  

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Contents

Group Theory
37
12 Problems about Groups
42
Some Refinements and Extensions
44
13 Dehn Functions
47
14 Group Homology
51
15 Arithmetic Groups
55
16 Realization of Sequences of Groups as Group Homology
60
Notes
63
32 Entropy of Free Loopspaces and Closed Contractible Geodesics
101
Constructing Aspherical Manifolds by Reflection Groups
105
33 Introduction to Kolmogorov Complexity
106
34 Complexity and Closed Geodesics
109
Notes
111
The LargeScale Fractal Geometry of Riemannian Moduli Space
119
42 Noeclassical Comparison Geometry
125
43 Existence of External Metrics
128

Designer Homology Spheres
69
22 Algorithmic Impossibility Results
73
23 Nabutovskys Thesis
76
24 The Classification of Homology Spheres
77
Surgery Homology Surgery and All That
80
Isotopy of Hypersurfaces
83
The Novikov Conjecture
85
25 Simplicial Norm
86
26 Homology Spheres with Nonzero Simplical Norm
88
Notes
90
The Roles of Entropy
97
44 Depth versus Density
131
45 BDiff
133
The Isomorphism Conjecture and Secondary Invariants
139
JSJ Decompositions
145
46 The Contagion of Symmetry
146
47 Filling Functions for RM
149
48 Further Directions
154
Notes
158
Index
171
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About the author (2005)

Shmuel Weinberger is Professor of Mathematics at the University of Chicago. He is the author of "The Topological Classification of Stratified Spaces.

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