Computers, Rigidity, and Moduli

by Weinberger, Shmuel

• ISBN13: 9780691118895
• ISBN10: 0691118892
• Format: Hardcover
• Copyright: 2004-11-29
• Publisher: Princeton Univ Pr

Summary

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 doppelgauml;ngers--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.

Table of Contents

Preface	ix
Introduction and Overview	1	(36)
I.1 Reflections on Light	1	(2)
I.2 Variational Problems	3	(2)
I.3 The Best Is Often Beautiful	5	(3)
I.4 Moduli Space (Phase Space)	8	(1)
I.5 Calculus and Beyond	9	(6)
I.6 Some Fine Print	15	(2)
I.7 The Limits of Computation (and of Proof)	17	(4)
I.8 And Beyond	21	(4)
I.9 The Method of Eastern Philosophy	25	(3)
I.10 Fractals and Geometricization	28	(4)
Notes	32	(5)
Chapter 1. Group Theory	37	(32)
1.1 Presentations of Groups	37	(5)
1.2 Problems about Groups	42	(5)
Appendix: Some Refinements and Extensions	44	(3)
1.3 Dehn Functions	47	(4)
1.4 Group Homology	51	(4)
1.5 Arithmetic Groups	55	(5)
1.6 Realization of Sequences of Groups as Group Homology	60	(3)
Notes	63	(6)
Chapter 2. Designer Homology Spheres	69	(28)
2.1 Fundamental Groups	69	(4)
2.2 Algorithmic Impossibility Results	73	(3)
2.3 Nabutovsky's Thesis	76	(1)
2.4 The Classification of Homology Spheres	77	(9)
Appendix 1: Surgery, Homology Surgery, and All That	80	(3)
Appendix 2: Isotopy of Hypersurfaces	83	(2)
Appendix 3: The Novikov Conjecture	85	(1)
2.5 Simplicial Norm	86	(2)
2.6 Homology Spheres with Nonzero Simplical Norm	88	(2)
Notes	90	(7)
Chapter 3. The Roles of Entropy	97	(22)
3.1 The Problem of Closed Geodesics	97	(4)
3.2 Entropy of Free Loopspaces and Closed Contractible Geodesics	101	(5)
Appendix: Constructing Aspherical Manifolds by Reflection Groups	105	(1)
3.3 Introduction to Kolmogorov Complexity	106	(3)
3.4 Complexity and Closed Geodesics	109	(2)
Notes	111	(8)
Chapter 4. The Large-Scale Fractal Geometry of Riemannian Moduli Space	119	(52)
4.1 Statement of Results	119	(6)
4.2 Neoclassical Comparison Geometry	125	(3)
4.3 Existence of Extremal Metrics	128	(3)
4.4 Depth versus Density	131	(2)
4.5 BDiff	133	(13)
Appendix 1: The Isomorphism Conjecture and Secondary Invariants	139	(6)
Appendix 2: JSJ Decompositions	145	(1)
4.6 The Contagion of Symmetry	146	(3)
4.7 Filling Functions for R(M)	149	(5)
4.8 Further Directions	154	(4)
Notes	158	(13)
Index	171