Hilbert’s fifth problem and related topics.
Terence Tao

Graduate Studies in Mathematics, Vol. 153

American Mathematical Society, 2014

338 pp., hardcover

ISBN-10: 1-4704-1564-X
ISBN-13: 978-1-4704-1564-8

Last updated: Apr 18, 2019

This continues my series of books derived from my blog, and is based on the lecture notes for my graduate course of the same name. The preceding books in this series were “Structure and Randomness“, “Poincaré’s legacies“, “An epsilon of room“, “An introduction to measure theory“, “Topics in random matrix theory“, “Higher order Fourier analysis“, and “Compactness and contradiction“.

An online version of the MS can be found here.

This book received the 2015 PROSE award in the mathematics category.

Pre-errata (to be corrected in the published version):

  • Page 5: In the proof of Theorem 1.1.2, \varepsilon should be taken to be 1/10n, rather than 1/10d.
  • Page 10: In Exercise 1.1.7, (x+1,0) should be (x+1,1), and (x,1) should be (x,0).
  • Page 20: In Problem 1.1.14, the hypothesis that G has polynomial growth is missing and should be inserted.
  • Page 79: In the last paragraph, a right parenthesis is missing after “Exercise 1.4.3”.


  • Page 16: In Example 1.2.5, the series x - x^2 + x^3 -\dots should be -x + x^2 - x^3 + \dots.
  • Page 18: In Definition 1.2.8, all occurrences of G should be replaced with M.
  • Page 54: In Exercise 3.0.8, G should be assumed to be Hausdorff.
  • Page 105: After (5.9), “\|g^n\| \leq \varepsilon and so g^n \in B(0,\varepsilon)” should be “for all m \leq n, \|g^n\| \leq \varepsilon and so g^n \in B(0,\varepsilon)” and similarly for (g^y)^n \in B(0,5\varepsilon).
  • Page 109: After (5.13), a “the” is missing in “takes values in [0,1] obeys the Lipschitz bound”.
  • Page 117: In the proof of Proposition 5.5.1: “\varepsilon is small enough” should be “U_1, B, \varepsilon are small enough”.
  • Page 181: In remark 8.2.3, it should be added that in the more general non-symmetric case discussed here, A^4 needs to be replaced by A^2 \cdot (A^{-1})^2.   Also, in the analogue of Exercise 8.2.2 in this more general case, A^2 needs to be replaced by an arbitrary translate g \cdot A of A.
  • Page 302: In most of this section \rho_\xi(g) should be \rho_\xi(g)^* and vice versa (also the subscript of \rho by \xi is missing in several places).

Thanks to Michael Cowling, Frederik vom Ende, Mikhail Katz, Lam Pham, Arturo Rodríguez Fanlo, Fan Zheng, and an anonymous contributor for corrections.