There are many bits and pieces of folklore in mathematics that are passed down from advisor to student, or from collaborator to collaborator, but which are too fuzzy and nonrigorous to be discussed in the formal literature. Traditionally, it was a matter of luck and location as to who learned such "folklore mathematics". But today, such bits and pieces can be communicated effectively and efficiently via the semiformal medium of research blogging. This book grew from such a blog. The articles, essays, and notes in this book are derived from the author's mathematical blog in 2010. It contains a broad selection of mathematical expositions, commentary, and self-contained technical notes in many areas of mathematics, such as logic, group theory, analysis, and partial differential equations. The topics range from the foundations of mathematics to discussions of recent mathematical breakthroughs. Lecture notes from the author's courses that appeared on the blog have been published separately in the Graduate Studies in Mathematics series.
Author(s): Terence Tao
Edition: Draft ed
Publisher: American Mathematical Society
Year: 2013
Language: English
Commentary: From: https://terrytao.wordpress.com/books/compactness-and-contradiction/
Pages: 262
Tags: Analysis; Mathematical Logic; Group Theory; Partial Differential Equations
Preface xi
A remark on notation xi
Acknowledgments xii
Chapter 1. Logic and foundations 1
§1.1. Material implication 1
§1.2. Errors in mathematical proofs 2
§1.3. Mathematical strength 4
§1.4. Stable implications 6
§1.5. Notational conventions 8
§1.6. Abstraction 9
§1.7. Circular arguments 11
§1.8. The classical number systems 12
§1.9. Round numbers 15
§1.10. The “no self-defeating object” argument, revisited 16
§1.11. The “no self-defeating object” argument, and the vagueness paradox 28
§1.12. A computational perspective on set theory 35
Chapter 2. Group theory 51
§2.1. Torsors 51
§2.2. Active and passive transformations 54
§2.3. Cayley graphs and the geometry of groups 56
§2.4. Group extensions 62
§2.5. A proof of Gromov’s theorem 69
Chapter 3. Analysis 79
§3.1. Orders of magnitude, and tropical geometry 79
§3.2. Descriptive set theory vs. Lebesgue set theory 81
§3.3. Complex analysis vs. real analysis 82
§3.4. Sharp inequalities 85
§3.5. Implied constants and asymptotic notation 87
§3.6. Brownian snowflakes 88
§3.7. The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation 88
§3.8. Finitary consequences of the invariant subspace problem 104
§3.9. The Guth-Katz result on the Erdös distance problem 110
§3.10. The Bourgain-Guth method for proving restriction theorems 123
Chapter 4. Nonstandard analysis 133
§4.1. Real numbers, nonstandard real numbers, and finite precision arithmetic 133
§4.2. Nonstandard analysis as algebraic analysis 135
§4.3. Compactness and contradiction: the correspondence principle in ergodic theory 137
§4.4. Nonstandard analysis as a completion of standard analysis 150
§4.5. Concentration compactness via nonstandard analysis 167
Chapter 5. Partial differential equations 181
§5.1. Quasilinear well-posedness 181
§5.2. A type diagram for function spaces 189
§5.3. Amplitude-frequency dynamics for semilinear dispersive equations 194
§5.4. The Euler-Arnold equation 203
Chapter 6. Miscellaneous 217
§6.1. Multiplicity of perspective 218
§6.2. Memorisation vs. derivation 220
§6.3. Coordinates 223
§6.4. Spatial scales 227
§6.5. Averaging 229
§6.6. What colour is the sun? 231
§6.7. Zeno’s paradoxes and induction 233
§6.8. Jevons’ paradox 234
§6.9. Bayesian probability 237
§6.10. Best, worst, and average-case analysis 242
§6.11. Duality 245
§6.12. Open and closed conditions 247
Bibliography 249
Index 255
