Last updated: Jan 22, 2012

Poincaré’s legacies: pages from year two of a mathematical blog (Volume I, Volume II)
Terence Tao
American Mathematical Society
Volume I: ISBN-10 0-8218-4883-6, ISBN-13 978-0-8218-4883-8
Volume II: ISBN-10 0-8218-4885-2, ISBN-13 978-0-8218-4885-2

This is a sequel to “Structure and Randomness: Pages from year one of a mathematical blog“, in two volumes.

A draft version of the MS can be found here (note that the printed version will have substantially different page and section numbering, in particular being split into two volumes).

The front cover for the first volume is here, and for the second volume is here.

See also this blog announcement.

– Errata for the first volume –

  • Page 21: In the first display after (1.19), \sum_{j=1}^\infty \frac{1}{n_j} X_{\leq n_j} should be \sum_{j=1}^\infty \frac{1}{n_j} |X_{\leq n_j}|^2.
  • Page 27: In (1.37), |t| should be |t|_p.
  • Page 64: In the two long displays the symbol P is missing just before the right bracket ] on most of the lines of the displays.
  • Page 87: In Exercise 2.2.4, the last sentence should be phrased as a question, i.e. “Does there exist analogous claims in the categories of dynamical systems and measure-preserving systems?”.
  • Page 110: A similar ultrafilter proof also appears in Section 3 of N. Hindman’s paper “Problems and new results in the algebra of Beta S and Ramsey Theory” in “Unsolved problems on mathematics for the 21st century”, J. Abe and S. Tanaka eds., IOS Press, Amsterdam (2001), 295-305.
  • Page 134: In Example 2.7.2, (0,1/2n) should be (1/2n,0), and (\alpha, \frac{n(n-1)}{2} \alpha + \frac{1}{2}) should be (n\alpha, \frac{n(n-1)}{2} \alpha + \frac{1}{2}).
  • Page 141: Exercise 2.7.14 is the same as 2.9.13 and should be deleted.
  • Page 143:  The last sentence of the proof of Theorem 2.8.2 is redundant and should be deleted.  In Exercise 2.8.3, \mu(X) should read \mu(E) (two occurrences).
  • Page 144: The first proof of von Neumann’s ergodic theorem is due to F. Riesz, rather than von Neumann, and the text should be edited accordingly.
  • Page 159: In Exercise 2.9.13, one needs to add the additional hypothesis that the support of the invariant measure \mu is equal to the whole space X.
  • Page 162: In the right-hand side of (2.96), the factor g(y) should be moved outside the inner integral (for clarity).  In Exercise 2.9.14, \nu_y should be \mu_y.
  • Pages 189, 194: In Exercise 2.12.15, and also in the first paragraph of Section2.12.4, Corollary 2.12.8 should be Corollary 2.12.13.
  • Page 210: In Proposition 2.14.11, the “weak operator topology” should be clarified to “the weak operator topology of L^2(X)“, and it should also be parenthetically noted that the S_{f,N} are uniformly bounded in the Hilbert space L^2(X).
  • Page 218: In Exercise 2.16.1(7), “H/[H,K] and K/[H,K] become abelian” should be “the images of H and K become groups that commute with each other”.
  • Page 221: In Example 2.16.9, [0,y+x \hbox{ mod } 1] should just be [0,y].
  • Page 222: In Example 2.16.13, the group element g should have a coefficient of -1 instead of 1 in the third column, second row position.
  • Page 223: In (2.203), n+1 should be n-1.
  • Page 231: The proof of Lemma 2.17.5 is incomplete, because U and D do not fully generate SL_2({\bf R}).  To finish the argument, observe that d^t w^\varepsilon d^{-t} converges to the identity as t \to +\infty, and thus \langle \rho(d^t w^\varepsilon d^{-t}) v, v \rangle \to \langle v, v \rangle.  Using the D-invariance we conclude that \rho(w^\varepsilon) v, v \rangle = \langle v, v \rangle, and thus as before v is also invariant with respect to the group U’ generated by the w^\varepsilon.  Since U and U’ (and D, if desired) generate SL_2({\Bbb R}), the claim follows.
  • Pages 232-233: The proof of Lemma 2.17.9 requires some changes.  In the penultimate paragraph, “any g in L” should be “any g in L with gx_0 sufficiently close to x_0“.    The final paragraph needs to be changed to the following: “Suppose that Lx_0 is not closed; then one can find a sequence g_n x_0 in Lx_0 that converges to x_0 but with the g_m g_n^{-1} staying bounded away from the identity for m \neq n.  For a sufficiently small compact neighbourhood K of the identity in L, the sets K g_n x_0 then are disjoint and all have the same measure for n large enough; since \mu(Lx_0)=1, this forces these sets to be null.  But then the invariant measure m annihilates K and is thus null as well, a contradiction.”

– Errata for the second volume –

  • Page 136: After (2.170), “slows down the flow of time by 1/\lambda” should be “slows down the flow of time by 1/\lambda^2“.
  • Page 229: The formulation of the Hamilton compactness theorem given here needs an additional hypothesis, namely a uniform lower bound on the Ricci curvature.  More precisely, for any compact interval J there exists a K such that for every radius r one has \hbox{Ric} \geq -K on J \times B_{g_n(t_0)}(p_n,r) for all sufficiently large n.  This is needed to prevent the length of a geodesic going off to infinity from collapsing to a finite length, causing incompleteness.  (It was recently shown by Topping that the formulation of the compactness theorem give in the text can fail without such a hypothesis.  However, in the applications to the Poincare conjecture one has the uniform lower bound on curvature, so this is ultimately not a major issue.)
  • Page 270: “width of the necks goes to infinity” should be “width of the necks goes to zero”.
  • Page 290: The reference [Zhang2007] should be “Zhang, Qi S.,   Strong noncollapsing and uniform Sobolev inequalities for Ricci flow with surgeries. Pacific J. Math. 239 (2009), no. 1, 179–200″.

Thanks to Paul-Olivier Dehaye, Neil Hindman, Ioannis Kontoyiannis, Sajjad Lakzian, Xiaochuan Liu, Hee Oh, Pavel, Robert Tu, Siming Tu, Mate Wierdl, Qi Zhang, Tamar Ziegler, Pavel Zorin, and an anonymous commenter for corrections and references.