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 American Mathematical Society, I published many of the mathematical articles from the first two years of the blog as [Ta2008] and [Ta2009], which will hence- forth be referred to as Structure and Randomness and Poincar´ e’s Legacies Vols. I, II 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 by the readers of the blog. The current text contains many (though not all) of the posts for the third year (2009) of the blog, focusing primarily on those posts of a mathematical nature which were not contributed primarily by other authors, and which are not published elsewhere. It has been split into two volumes. The first volume (referred to henceforth as Volume 1 ) consisted primarily of lecture notes from my graduate courses on real analysis that I taught at UCLA. The current volume consists instead of sundry articles on a variety of mathematical topics, which I have divided (somewhat arbitrarily) into expository articles (Chapter 1) which are introductory articles on topics of relatively broad interest, and more technical articles (Chapter 2) which are narrower in scope and often related to one of my current research interests. vii
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