2009;
293 pp;
Softcover

MSC: Primary 00;

**Print ISBN: 978-0-8218-4883-8
Product Code: MBK/66**

List Price: $44.00

AMS Member Price: $35.20

MAA Member Price: $39.60

**Electronic ISBN: 978-1-4704-1601-0
Product Code: MBK/66.E**

List Price: $41.00

AMS Member Price: $32.80

MAA Member Price: $36.90

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#### Supplemental Materials

# Poincaré’s Legacies, Part I: pages from year two of a mathematical blog

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*Terence Tao*

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 to cover a variety of
topics, ranging from his own research and other recent developments in
mathematics, to lecture notes for his classes, to non-technical puzzles
and expository articles. The articles from the first year of that blog
have already been published by the AMS. The posts from 2008
are being published in two volumes.

This book is Part I of the second-year posts, focusing on ergodic
theory, combinatorics, and number theory. Chapter 2 consists of
lecture notes from Tao's course on topological dynamics and ergodic
theory. By means of various correspondence principles, recurrence
theorems about dynamical systems are used to prove some deep theorems
in combinatorics and other areas of mathematics. The lectures are as
self-contained as possible, focusing more on the “big
picture” than on technical details.

In addition to these lectures, a variety of other topics are discussed,
ranging from recent developments in additive prime number theory to
expository articles on individual mathematical topics such as the law of
large numbers and the Lucas–Lehmer test for Mersenne primes. Some
selected comments and feedback from blog readers have also been
incorporated into the articles.

The book is suitable for graduate students and research mathematicians
interested in broad exposure to mathematical topics.

#### Readership

Graduate students and research mathematicians interested in mathematics in general with a focus on ergodic theory, combinatorics, and number theory.

#### Reviews & Endorsements

Tao's mathematical knowledge has an extraordinary combination of breadth and depth: he can write confidently and authoritatively on topics ... Reading these extended discussions in book form will, for many people at least, be easier than reading them on the blog.

-- Mathematical Reviews

[This book] is demanding, entertaining, provides you with the big picture behind the sometimes technical results, and certainly gives you your money's worth. ... Armed with a minimal background in number theory, these lectures can be read with profit by advanced undergraduates.

-- Zentralblatt MATH