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
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