Structure and Randomness: pages from year one of a mathematical blog

American Mathematical Society, 2008

298 pages

ISBN-10: 0-8218-4695-7

ISBN-13: 978-0-8218-4695-7

Last updated Sep 26, 2020

This is a book version of my blog, covering many (though not all) of my articles in 2007, reworked into a publishable format (and in particular, with formal and updated references).

A draft copy of the book is available here. Note that the formatting for this internet version is significantly different from that in the final print version, in particular the page numbering does not correspond at all.

Here is a (2MB) PDF file containing the front cover of the book.

The official AMS web page for this book is here.

A review of the book for the American Scientist is here.

Here is a review for the MAA.

— Errata —

- Page 21: In Proposition 1.8, should be .
- Page 22: In Proposition 1.10, should be .Page 51: In Lemma 1.34(2), should be .
- Page 54: The phrase “nonstandard ultrafilter on ” requires some clarification. It should be “ultrafilter on that extends an ultraproduct of nonprincipal ultrafilters on “.
- Page 86: In the final display, the exponent of on the LHS should be deleted. Similarly on the first display of page 87.
- Page 95, Figure 1: The graph of is missing two edges (namely, the long diagonal edges between opposite corners of the graph).
- Page 96: The inequality is only valid for connected graphs with at least one cycle. In general, one only has . One must then delete all terms involving the number 6 in the rest of the section. One can then replace “To solve the optimisation problem exactly, one needs to solve a cubic; but we can perform a much cheaper computation” by “One can solve the optimisation problem exactly, but we can perform a cheaper computation”.
- Page 101: Footnote 67 is incorrect and should be deleted.
- Page 102: should be .
- Page 103: The sentence “More generally, any Riemannian manifold…” is incorrect and should be deleted, replaced instead with the sentence “Similarly, the hy0erbolic plane is isomorphic to “.
- Page 104: In first paragraph: the modular curve should be . In the footnote, add “and an obvious left action of ,” after the final comma.
- Page 128: In the second and third displays, should be .
- Page 129: “decaying faster” should be “decaying faster or having smaller amplitude”.
- Page 143: The proof that is not correct as stated (the perturbations indicated do not preserve the property of being an -flow). A correct argument is as follows. Call an edge of an -flow
*unsaturated*if it has weight strictly between 0 and , and similarly call a vertex*unsaturated*if its net inflow or outflow is strictly less than . Observe that if e is an unsaturated edge, then the final vertex u of e will either have an unsaturated edge leading out of it (if u is unsaturated) or another unsaturated edge leading into it (if u is saturated). Similarly, the initial vertex u’ of e will either have an unsaturated edge leading into it (if u is unsaturated) or another unsaturated edge leading out of it (if u is saturated). Thus, if there is at least one unsaturated edge, then by iterating the above observations, one can find an oriented cycle along unsaturated edges with the property that at any saturated vertex u, the number of edges flowing along the cycle into u equals the number of edges flowing against the cycle into u, and the number of edges flowing along the cycle out of u equals the number of edges flowing against the cycle out of u. For this cycle, one can modify the flow as indicated in the text to reduce the number of edges in the -flow. - Page 137: In the last line of Case 1, “multiply (1.47) by…” should be “raise (1.47) to the power and then multiply by…”, and should be .
- Page 148: In the first line, should read .
- Page 180: In the second display, should be .
- Page 257: When selecting the large prime , it is necessary that is larger than 2. This is needed in order for the binomial expansion of to be expressible in the desired form where the polynomials have coefficients in the p-adic integers, because the number of times divides can be bounded below by , which goes to infinity as provided that is larger than 2.
- Page 267: In the third display, should be .
- Section 3.10: In (2.8), should be .

Thanks to Ravindra Bapat, Alan Chang, Dion, Norman Hardy, Matthew Kahle, Chris Kimmel, JamesL, Avi Levy, Russ Lyons, Arturo Magadin, Kirane Mokhtar, David P. Moulton, Kamil Rychlewicz , David Speyer, Nikodem Szpak, Tobias and Tony, Chao Weng, Sheng-Peng Wu, and Yuncheng for corrections.

## 11 comments

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4 February, 2008 at 6:00 am

Prokrastination oder Blogroll (I): Terence Tao at LEMUREN-Blog[…] Lässt man sich von den grösstenteils mathematisch anspruchsvollen Beiträgen, welche Tao nun in einem Buch zusammenfassen und veröffentlichen wird, so findet man dort neben Hinweisen zu “Career-Advice” und “On […]

6 August, 2008 at 6:02 am

the future of this blog | neverendingbooks[…] I can see the point in setting up a blog connected to a book you once wrote or intend to write (such as Not Even Wrong or Terry Tao). […]

23 November, 2008 at 2:41 pm

Structure and Randomness: pages from year one of mathematical blog « What’s new[…] AMS has just notified me that the book version of the first year of my blog, now retitled “Structure and Randomness: pages from year one of a mathematical blog“, is now available. An official web page for this book has also been set up here, though it […]

29 January, 2009 at 10:27 am

Poincaré’s legacies: pages from year two of a mathematical blog « What’s new[…] in the previous book, those comments and corrections from readers which were of a substantive and mathematical nature […]

28 August, 2009 at 11:37 am

» Science 2.0: What Every Scientist Needs to Know About How the Web is Changing the Way They Work Information/Science[…] Tao’s blog is professional enough that it has been formally published in two volumes: “Structure and Randomness: pages from year one of a mathematical blog” and “Poincaré’s legacies: pages from year two of a mathematical […]

8 February, 2010 at 2:27 pm

An epsilon of room: pages from year three of a mathematical blog « What’s new[…] a mathematical blog“. It largely follows the format of my previous two blog books, “Structure and Randomness“ and “Poincaré’s […]

2 May, 2012 at 7:32 am

Arturo MagidinI believe there is an error on page 95, Figure 1. The graph labeled as $K_{3,3}$ is not $K_{3,3}$, since the corner vertices are not connected to the opposite corners.

[Correction added, thanks – T.]2 August, 2013 at 6:18 pm

MattI recently purchased this book in e-form through Google books – I was surprised that unlike the beautifully published print version, which I also own, Google appears to be selling a rather low-quality scanned version of your book, wrinkles visible, which is not searchable, text-reflow-able, nor available to read when not plugged into the internet. If I know mathematicians, I know that most would be appalled that their work was being presented, even sold, in such degraded form. Would you be kind enough to contact your publisher, AMS, and urge them to supply EPUB versions of your books to Google? This would be greatly appreciated, and I very much look forward to reading your articles on the subway without needing a wheelbarrow to carry the all the dead trees :) Thanks!

26 May, 2016 at 8:55 am

KerriStructure and randomness : where a simple law explodes into quantum elements of order that describe the bounds of physics. Where space is as important in understanding the order of structures as energy and matter is. These concepts are Real within Prime structure and pi, some of our favorite explorations of infinite possibility.

13 November, 2017 at 12:48 pm

AnonymousThere are a couple errors on pages 21-22. Propositions 1.8 and 1.10 are not correctly stated; a counterexample to both is

\epsilon = 4/5

k = 2

M = 4

x_1 = 0, x_2 = 0.1, x_3 = 0.9, x_4 = 1

The bound needed on M in Proposition 1.8 is stronger: $\lfloor\frac{M-1}{k}\rfloor \geq \epsilon$. Along the same lines, the bound needed on M in Proposition 1.10 is $\lfloor\log_2 (M) \rfloor$.

With these modifications, the proofs alluded to after each proposition do work. I suggest also changing $x_k, x_2k, x_3k, …$ to $x_1, x_{1+k}, x_{1+2k}, …$ on page 21 and removing the variable k from Proposition 1.10.

Thank you for the excellent article; I’m still working through it.

[Errata added, thanks – T.]2 October, 2019 at 5:16 am

rongjiang panA Proof of Collatz Conjecture in Binary

https://github.com/ronaldpan/Collatz