The Plaid Model

Name: The Plaid Model | (AMS-198)
Brand: Princeton University Press
Price: 391.95 EUR
Availability: OnlineOnly

(AMS-198)

Richard Evan Schwartz(Autor*in)

Princeton University Press

1. Auflage

Erschienen am 19. Februar 2019

280 Seiten

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978-0-691-18899-7 (ISBN)

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Inhalt

Cover
Title
Copyright
Contents
Preface
Introduction
0.1 Part 1: The Plaid Model and its Properties
0.2 Part 2: The Plaid PET
0.3 Part 3: The Graph PET
0.4 Part 4: Plaid-Graph Correspondence
0.5 Part 5: The Distribution of Orbits
0.6 Companion Program
PART 1. THE PLAID MODEL
Chapter 1. Definition of the Plaid Model
1.1 Chapter Overview
1.2 Basic Quantities and Notation
1.3 Six Families of Lines
1.4 Capacity, Mass, and Sign
1.5 Light Points
1.6 Transverse Directions for the Light Points
1.7 Main Definition
Chapter 2. Properties of the Model
2.1 Chapter Overview
2.2 Symmetries
2.3 The Number of Intersection Points
2.4 The Meaning of Capacity
2.5 A Subtle Symmetry
Chapter 3. Using the Model
3.1 Chapter Overview
3.2 The Big Polygon
3.3 Hierarchical Information
3.4 A Subdivision Algorithm
3.5 Comparing Different Parameters
Chapter 4. Particles and Spacetime Diagrams
4.1 Chapter Overview
4.2 Remote Adjacency
4.3 Horizontal Particles
4.4 Vertical Particles
4.5 Spacetime Diagrams and Their Symmetries
4.6 The Bad Tile Lemma
Chapter 5. Three-Dimensional Interpretation
5.1 Chapter Overview
5.2 Stacking the Blocks
5.3 Pixelated Spacetime Diagrams
5.4 Tile Compatibility
5.5 Spacetime Plaid Surfaces
5.6 Discussion and Speculation
Chapter 6. Pixellation and Curve Turning
6.1 Chapter Overview
6.2 Orienting the Worldlines
6.3 The Sparseness of Worldlines
6.4 Curve Turning Theorem: Vertical Case
6.5 Curve Turning Theorem: Horizontal Case
6.6 Two Applications
Chapter 7. Connection to the Truchet Tile System
7.1 Chapter Overview
7.2 Truchet Tilings
7.3 The Truchet Comparison Theorem
7.4 The Fundamental Surface
7.5 A Result from Elementary Number Theory
7.6 Proof of the Truchet Comparison Theorem
PART 2. THE PLAID PET
Chapter 8. The Plaid Master Picture Theorem
8.1 Chapter Overview
8.2 The Spaces
8.3 The Checkerboard Partition
8.4 The Classifying Map
8.5 The Main Result
Chapter 9. The Segment Lemma
9.1 Chapter Overview
9.2 The Anchor Point
9.3 A Computational Tool
9.4 The Vertical Case
9.5 The Horizontal Case
Chapter 10. The Vertical Lemma
10.1 Chapter Overview
10.2 Using Symmetry
10.3 Translating the Picture
10.4 Some Useful Formulas
10.5 The Undirected Result
10.6 Determining the Directions
Chapter 11. The Horizontal Lemma
11.1 Chapter Overview
11.2 Using Symmetry
11.3 Translating the Picture
11.4 Two Easy Technical Lemmas
11.5 The Undirected Result
11.6 Determining the Directions
Chapter 12. Proof of the Main Result
12.1 Chapter Overview
12.2 Prism Structure
12.3 Some Extra Symmetry
12.4 The Vertical Case
12.5 The Horizontal Case
PART 3. THE GRAPH PET
Chapter 13. Graph Master Picture Theorem
13.1 Chapter Overview
13.2 Special Orbits
13.3 The Arithmetic Graph
13.4 A Preliminary Result
13.5 The PET Structure
13.6 The Fundamental Polytopes
Chapter 14. Pinwheels and Quarter Turns
14.1 Chapter Overview
14.2 The Pinwheel Map
14.3 Outer Billiards and the Pinwheel Map
14.4 Quarter Turn Compositions
14.5 The Pinwheel Map as a QTC
14.6 The Case of Kites
Chapter 15. Quarter Turn Compositions and PETs
15.1 Chapter Overview
15.2 A Result from Linear Algebra
15.3 The Map
15.4 Compactifying Shears
15.5 Compactifying Quarter Turn Maps
15.6 The End of the Proof
Chapter 16. The Nature of the Compactification
16.1 Chaper Overview
16.2 The Singular Directions
16.3 The First Parallelotope
16.4 The Second Parallelotope
16.5 The General Master Picture Theorem
16.6 Structure of the PET
16.7 The Case of Kites
PART 4. THE PLAID-GRAPH CORRESPONDENCE
Chapter 17. The Orbit Equivalence Theorem
17.1 Chapter Overview
17.2 The Prisms
17.3 The Map
17.4 Characterizing the Image
17.5 The Clean Partition
17.6 The Main Proof
17.7 Computational Techniques
17.8 The Calculations
Chapter 18. The Quasi-Isomorphism Theorem
18.1 Chapter Overview
18.2 The Canonical Affine Transformation
18.3 The Graph Grid
18.4 The Intertwining Lemma
18.5 The Correspondence of Orbits
18.6 The End of the Proof
18.7 The Projection Theorem
18.8 Renormalization Interpretation
Chapter 19. Geometry of the Graph Grid
19.1 Chapter Overview
19.2 The Grid Geometry Lemma
19.3 The Graph Reconstruction Lemma
Chapter 20. The Intertwining Lemma
20.1 Chapter Overview
20.2 A Resume of Transformations
20.3 Injectivity of the Map
20.4 Calculating a Single Point
20.5 Dissecting the Set
20.6 The Induction Step
20.7 Discussion
20.8 The Diagonal Case
PART 5. THE DISTRIBUTION OF ORBITS
Chapter 21. Existence of Infinite Orbits
21.1 Chapter Overview
21.2 Definedness Criterion
21.3 Spacetime Diagrams Revisited
21.4 Taking a Limit
21.5 Associated Paths
21.6 Sketch of an Alternate Proof
Chapter 22. Existence of Many Large Orbits
22.1 Chapter Overview
22.2 Equidistribution Properties
22.3 The Ubiquity Lemma
22.4 The Rectangle Lemma
22.5 Proof of the Main Result
22.6 The Continued Fraction Length
22.7 The End of the Proof
Chapter 23. Infinite Orbits Revisited
23.1 Chapter Overview
23.2 The Approximating Sequence
23.3 The Copy Theorem
23.4 The End of the Proof
23.5 The Copy Lemma
23.6 Proof of the Box Theorem
23.7 Proof of the Copy Theorem
23.8 Hidden Symmetries
Chapter 24. Some Elementary Number Theory
24.1 Chapter Overview
24.2 A Structural Result
24.3 Unfinished Business
Chapter 25. The Weak and Strong Case
25.1 Chapter Overview
25.2 The First Two Statements
25.3 A Technical Lemma
25.4 The Mass and Capacity Sequences
25.5 Vertical Intersection Points
25.6 A Matching Criterion
25.7 Verifying the Matching Criterion
Chapter 26. The Core Case
26.1 Chapter Overview
26.2 The First Two Statements
26.3 Geometric and Arithmetic Alignment
26.4 Geometric Alignment
26.5 Alignment of the Capacity Sequences
26.6 A Technical Lemma
26.7 The Mass Sequences: Central Case
26.8 The Mass Sequences: Peripheral Case
26.9 The End of the Proof
Appendix References
Index

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