Preface In February of 2007, I converted my “What’s new” web page of research updates into a blog at 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 vii
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