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Poincaré’s legacies. Part II. Pages from year two of a mathematical blog. (English) Zbl 1175.00010

Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4885-2/pbk). ix, 292 p. (2009).
Part II of this volume (see Zbl 1171.00003 for Part I) containing updated and corrected versions of the second year of the author’s blog on mathematics has two parts. The first discusses various topics such as Kakeya’s needle problem on the minimal area of a region required for rotating a needle (this theme appears three times), the Black-Scholes equation, gauge theories, the stability of eigenvalues, and others.
The major part of this book consists of lecture notes on Perelman’s solution of the Poincaré conjecture. It starts by reviewing material from Riemannian geometry (curvature, flows etc.), and then explain’s Perelman’s approach to the Poincaré conjecture via Ricci flows. The next few sections deal with results on the extinction of various homotopy groups, and then the properties of the Ricci flow needed to prove the conjecture are derived. The author does his best to motivate the constructions, he gives outlines of certain proofs, and often explains why more simple-minded approaches do not seem to work.
It should be rather obvious that a lot of background is required for reading this book; on the other hand, the entertaining style of writing makes this book fun to read even for those who only have a basic command in Riemannian geometry or algebraic topology etc.

MSC:

00A05 Mathematics in general
00B15 Collections of articles of miscellaneous specific interest
37-02 Research exposition (monographs, survey articles) pertaining to dynamical systems and ergodic theory
37A99 Ergodic theory
11A41 Primes

Citations:

Zbl 1171.00003
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