Preface

In February of 2007, I converted my “What’s new” web page of research

updates into a blog at terrytao.wordpress.com. This blog has since grown

and evolved to cover a wide variety of mathematical topics, ranging from my

own research updates, to lectures and guest posts by other mathematicians,

to open problems, to class lecture notes, to expository articles at both basic

and advanced levels.

With the encouragement of my blog readers, and also of the AMS, I

published many of the mathematical articles from the first year (2007) of

the blog as [Ta2008b], which will henceforth be referred to as Structure

and Randomness throughout this book. This gave me the opportunity to

improve and update these articles to a publishable (and citeable) standard,

and also to record some of the substantive feedback I had received on these

articles from the readers of the blog. Given the success of the blog experi-

ment so far, I am now doing the same for the second year (2008) of articles

from the blog. This year, the amount of material is large enough that the

blog will be published in two volumes.

As with Structure and Randomness, each part begins with a collection of

expository articles, ranging in level from completely elementary logic puzzles

to remarks on recent research, which are only loosely related to each other

and to the rest of the book. However, in contrast to the previous book, the

bulk of these volumes is dominated by the lecture notes for two graduate

courses I gave during the year. The two courses stemmed from two very

different but fundamental contributions to mathematics by Henri Poincar´e,

which explains the title of the book.

This is the first of the two volumes, and it focuses on ergodic theory, com-

binatorics, and number theory. In particular, Chapter 2 contains the lecture

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