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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 nonrigorous 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. The articles, essays, and notes in this book are derived from the author's mathematical blog in 2010. It contains a broad selection of mathematical expositions, commentary, and self-contained technical notes in many areas of mathematics, such as logic, group theory, analysis, and partial differential equations. The topics range from the foundations of mathematics to discussions of recent mathematical breakthroughs. Lecture notes from the author's courses that appeared on the blog have been published separately in the Graduate Studies in Mathematics series.
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Terence Tao was the winner of the 2014 Breakthrough Prize in Mathematics. He is the James and Carol Collins Chair of mathematics at UCLA and the youngest person ever to be promoted to full professor at the age of 24. In 2006 Tao became the youngest ever mathematician to win the Fields Medal. His other honours include the George Polya Prize from the Society for Industrial and Applied Mathematics (2010), the Alan T Waterman Award from the National Science Foundation (2008), the SASTRA Ramanujan Prize (2006), the Clay Research Award from the Clay Mathematical Institute (2003), the Bocher Memorial Prize from the American Mathematical Society (2002) and the Salem Prize (2000).Review:
Lately, a new generation of experts, some of whom are very powerful researchers, has started disseminating exposition via interactive blogs. Since they receive immediate and continuous international feedback, the interests and needs of their audience inevitably inform the communication. Tao, who--even among Fields medalists--has exceptional intellectual range, writes among the most famous of these blogs, and the present volume contains a selection of redacted content from it. Whereas many textbooks aim to teach lower-division mathematics majors to prove trivial theorems, this book explores very generally how research mathematicians prove nontrivial theorems. But unifying themes aside, Tao has chosen very well as case studies sundry topics from mutually distant branches of mathematics. Those sufficiently elementary demand a healthy stretch, and none will seem overly familiar, even to most experts. The class exhortation "read the masters" takes on new urgency in an era when the masters return favor. Highly recommended ... [for] all academic and professional library collections. --D.V. Feldman, CHOICE
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