Structure and Randomness: Pages from Year One of a Mathematical Blog

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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 non-rigorous 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. In 2007, Terry Tao began a mathematical blog, as an outgrowth of his own website at UCLA. This book is based on a selection of articles from the first year of that blog. These articles discuss a wide range of mathematics and its applications, ranging from expository articles on quantum mechanics, Einstein's equation $E=mc^2$, or compressed sensing, to open problems in analysis, combinatorics, geometry, number theory, and algebra, to lecture series on random matrices, Fourier analysis, or the dichotomy between structure and randomness that is present in many subfields of mathematics, to more philosophical discussions on such topics as the interplay between finitary and infinitary in analysis. Some selected commentary from readers of the blog has also been included at the end of each article. While the articles vary widely in subject matter and level, they should be broadly accessible to readers with a general graduate mathematics background; the focus in many articles is on the ''big picture'' and on informal discussion, with technical details largely being left to the referenced literature.

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Product Details

  • ISBN-13: 9780821846957
  • Publisher: American Mathematical Society
  • Publication date: 11/18/2008
  • Pages: 298
  • Sales rank: 1,448,572
  • Product dimensions: 6.90 (w) x 9.90 (h) x 0.50 (d)

Table of Contents

Ch. 1 Expository Articles 1
1.1 Quantum mechanics and Tomb Raider 1
1.2 Compressed sensing and single-pixel cameras 10
1.3 Soft analysis, hard analysis, and the finite convergence principle 17
1.4 The Lebesgue differentiation theorem and the Szemeredi regularity lemma 30
1.5 Ultrafilters, non-standard analysis, and epsilon management 38
1.6 Dyadic models 57
1.7 "Math doesn't suck", and the Chayes-McKellar-Winn theorem 69
1.8 Nonfirstorderisability 79
1.9 Amplification, arbitrage, and the tensor power trick 83
1.10 The crossing number inequality 94
1.11 Ratner's theorems 103
1.12 Unipotent elements of the Lorentz group, and conic sections 110
1.13 The Jordan normal form and the Euclidean algorithm 118
1.14 John's blowup theorem for the non-linear wave equation 123
1.15 Hilbert's Nullstellensatz 130
1.16 The Hahn-Banach theorem, Menger's theorem, and Belly's theorem 139
1.17 Einstein's derivation of E = mc[superscript 2] 146 Ch. 2 Lectures 155
2.1 Simons Lecture Series: Structure and randomness 155
2.2 Ostrowski Lecture: The uniform uncertainty principle and compressed sensing 180
2.3 Milliman Lecture Series: Recent developments in arithmetic combinatorics 188 Ch. 3 Open Problems 211
3.1 Best bounds for cap sets 211
3.2 Non-commutative Freiman theorem 213
3.3 Mahler's conjecture for convex bodies 216
3.4 Why global regularity for Navier-Stokes is hard 220
3.5 Scarring for the Bunimovich stadium 234
3.6 Triangle and diamond densities in large dense graphs 239
3.7 What is a quantum honeycomb? 243
3.8 Boundedness of the trilinear Hilbert transform 249
3.9 Effective Skolem-Mahler-Lech theorem255
3.10 The parity problem in sieve theory 259
3.11 Deterministic RIP matrices 273
3.12 The non-linear Carleson conjecture 278 Bibliography 283
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