Books

This page is now obsolete.  The new version of the page can be found here.

The page below includes all available sample chapters and supplementary material for my published books.  I will not be able to respond to requests for full-length copies of the books.

Solving Mathematical Problems: A personal perspective
Terence Tao
(First edition) Deakin University Press, Geelong, Vic.: 1992
Hardcover. 85 pages. ISBN 0-7300-1365-0   
(Second edition) Oxford University Press, Oxford, England: 2006
Paper, 150 pages. ISBN13: 9780199205608
ISBN10: 0199205604

The first run of this book is unfortunately completely out of print (the entire press no longer exists).  It discusses various Olympiad level problems and how one can go about trying to solve them.  Oxford University Press has reprinting the book and it is now in print.

• Sample chapters (Preface, strategy, problems in number theory) 

Corrections to the second edition:

• In page 2, “Problem 1.1 question” should just be “Problem 1.1”.
• In page 7, “ompute” should be “compute”.  On the fourth displayed equation, 4d^2 should be 4d^4.
  
• In page 9, example (e), “876” should be “376”.
• In page 16, bottom, "217" should be "2^17".
• In page 25, third paragraph: one of the "n"s should be in math mode.
• In page 33, Exercise 2.5: For an additional challenge, prove this exercise without using Bertrand's postulate.
• In page 37, second display: f(f(2)-1) should be f(f(f(2)-1))).
• In page 44, 5ab should be (13)  (two occurrences)
• In page 45, Problem 3.4, there should be no commas between (x-a_1)^2 and (x-a_n)^2.  "are all integers" should be "are all distinct integers".
• In page 47, Exercise 3.7, the +1 should be a -1, one should look at p(x)+q(x) rather than p(x)-q(x), and "are integers" should be "are distinct integers".
• In the diagram on page 53, the angle at D should be gamma + beta/2, and the angle at E should be beta + gamma/2.
• In page 58, one of the instances of ABEF should be in math mode (like all the other instances).
• In page 66, problem 44, there is a "=2" missing at the end of the string of equations, thus x^p+y^q=y^r+z^p=z^q+x^r=2.
  
• In the diagrams on page 79 and page 82, the labels C and D should be switched.
• In page 95: change all occurrences of "rouble" to "ruble" (for consistency).
  

Thanks to Cecil Rousseau, Naoki Sato and Weiyu Xu for corrections.

Three Regularity Results in Harmonic Analysis
Terence Tao
in, Topics in analysis and its applications: selected theses
edited by R Coifman (Yale University)
World Scientific Publishing Co. 2000.
Hardcover, 464 pages. ISBN 981-02-4093-7
pp. 61-98

This is a mildly edited version of my thesis, collected together with the theses of Wilhelm Schlag, Christoph Thiele, Anna Gilbert, and Naoki Saito.

Analysis, Volume I
Analysis, Volume II
Terence Tao
Hindustan Book Agency, January 2006
Paper cover, 422 / 274 pages.  ISBN 81-85931-63-1 / 81-85931-62-3

This is basically an expanded and cleaned up version of my lecture notes for Math 131A and Math 131B.  In the US, it is available through the American Mathematical Society (Volume I and Volume II).

• Book I sample chapters (contents, natural numbers, set theory, integers and rationals, logic, decimal system, index)

Corrections to the printed version, Volume I:

• p.5, line 6 from bottom: sin(pi/2-2) should be sin(pi/2-z)
• p. 98: In Exercise 4,2,6, x,y,z should be rational numbers, not real.
• p. 101: In Definition 4.3.9, after "x^0 := 1", add "; in particular, we define 0^0 := 1".
• p. 127: In Exercise 5.3.4: add "(Hint: use Exercise 5.2.2.)".
• p. 131, line 12 from bottom: "they cannot be than" should be "they cannot be larger than".
• p. 175, Exercise 6.6.3: In the hint, replace "introduce" by "recursively introduce", and insert "; n > n_{j-1}" after "|a_n| >= j" (two occurrences), with the parenthetical "(omitting the n > n_{j-1} condition when j=0)" inserted after the recursive definition of n_j.
• p. 197, in second line of proof of Proposition 7.3.4: the second sum should be from k=0 to infinity, not from k=0 to K.
• p. 255, Proposition 9.3.9(b): f(x_0) should be L.
• p. 336, line 13: replace "we have made no assumption on alpha" with "the function alpha: R -> R could have been arbitrary".

• p. 392, example 12.1.7: 5+2=7 should be 3+4=7.
• p. 393, example 12.1.9: sup(5,2)=7 should be sup(3,4)=4.
• p. 394, example 12.1.13: (iii) should be (c).
• p. 403, Example 12.2.13: delete the redundant ", but not the other".
• p. 404, line 4: "neither open and closed" should be "neither open nor closed".
• p. 416, line 11: "k >= j" should be "k > j".
• p. 419, line -2: In Exercise 12.5.15, = should be \neq.  "that by counterexample" should be "by counterexample that"
• p. 426, Exercise 13.2.9: X should be R throughout.
  
• p. 452, Exercise 14.3.6: "Propositoin" should be "Proposition". 
  
• p. 452, Exercise 14.3.8: "x in R" should be "x in X".
• p. 458: Exercise 14.5.2 should be deleted and redirected to Exercise 14.2.2(c).
• p. 459: In line 11, 2 epsilon should be epsilon
  
• p. 464: ) missing at the end of Exercise 14.7.2.
• p. 466: Exercise 14.8.8 should be Exercise 14.8.2.
• p. 467: Exercise 14.8.11 should be Exercise 14.8.4.
• p. 469: "Limits of integration" should be "Limits of summation".  In Lemma 14.8.14, 3M+2\delta should be 1+4M, and Exercise 14.8.14 should be Exercise 14.8.6.
• p. 470: Exercise 14.8.15 should be Exercise 14.8.7.  Exercise 14.8.16 should refer to a (currently non-existent) Exe rcise 14.8.9, which of course would be to prove Lemma 14.8.16.
• p. 472: In Exercise 14.8.2(c), Lemma 14.8.2 should be Lemma 14.8.8.
• p. 486: In second and third display, y should be in [0,1] rather than (-1,1).
• p. 502: In Exercise 15.6.6, Lemma 15.6.6 should be Lemma 15.6.11.
• p. 556: In Theorem 17.5.4, f can take values in R^m and not just in R; insert the line "By working with one component of f at a time, we may assume m=1" as the first line of the proof.  f(x-x_0) should be f(x+x_0).
• p. 557: In the second display, -f(0) should be +f(0).
• p. 560: In Exercise 17.6.1, add the hypothesis "and f is continuous" before ", show that f is a strict contraction".
• p. 561: In Exercise 17.6.3, change "which is a strict contraction" to "such that |f(x)-f(y)| < |x-y| for all distinct x,y in [a,b]".  In Exercise 17.6.8, max(c,c') should be min(c,c').
• p. 565, line -7: U should be f^{-1}(B(0,r/2)) rather than f^{-1}(B(0,r)).
• p. 570, first display: all partial derivatives should have a - sign (not just the first one).
• p. 571, second display: add "=0" at the end.
  

Thanks to Manoranjan Majji, Cristina Pereyra, and the students of Math 401/501 and Math 402/502 at the University of New Mexico for corrections.

Additive Combinatorics
Terence Tao, Van Vu
Cambridge University Press
Hardback, 530 pages (ISBN-13: 9780521853866; ISBN-10: 0521853869)

This book covers the basic tools in additive combinatorics: sum set estimates, inverse theorems, graph theory techniques, crossing numbers, algebraic methods, Szemeredi’s theorem.

• Here are some sample chapters from the book. (Contents, probabilistic method, sum set estimates, additive geometry) 

And here are some “deleted scenes” that were cut from the book:

• Arithmetic Ramsey Theory
• Menger’s theorem
• Santalo’s inequality
• Entropy sumset estimates

And here are some expository notes which are precursors of material currently in the book:

• Gowers' proof of Szemeredi's theorem for progressions of length 4
• Non-commutative sum set estimates
• The Roth-Bourgain theorem
• Math 254A (some highlights of arithmetic combinatorics)
• Math 262 (Topics in Combinatorics)

Corrections to the printed version:

• P. 5: In Q. 1.1.5, A \cap (B+x) and A \cap (B+y) should be A \cup (B+x) and A \cup (B+y) respectively.
• P. 8: In the fourth display, \ln should be \log.
• P. 11: In the paragraph after (1.17), ln should be log.
  
• P. 14: In the 9th line from below, ln should be log (two places); similarly before (1.23) on P. 15.
• P. 23: In the last line of the proof of Lemma 1.24, \ln should be \log.
  
• P. 68: In Exercise 2.3.14, A-A should be A+A and x+A should be x-A.  Replace “though the set F may be symmetric around a non-zero origin x/2” with “where by symmetry here we mean that F=x-F”.  (The point is that x/2 might not actually exist and be unique for some groups.)
• P. 79: In (2.20), K^4 should be K^5. 
  
• P. 95: Lemma 2.41, while correct, does not imply Corollary 2.42 as stated (since d(A,A) does not control |A^{-1} . A|).  Thanks to Mei-Chu Chang for pointing out the problem.  Replace “As d(A,A) <= 2d(A,B), this implies” by “A similar argument (see [Proposition 4.5, 362]) gives”.  Rename Corollary 2.42 as Proposition 2.42, replace “B” by “A” in the first sentence, replace “|S| >= Omega(K^{-O(1)} |A|}” by “|S| >= |A|/2K” in the second sentence,  and replace “O(K)^{O(1+n)}” by “2^n K^{2n+1}” in the display.
• P. 96: “verstion” should be “version”.
• P. 107: before Corollary 2.62, "Corollary 2.60" should be "Theorem 2.60".
  
• P. 141-142: Throughout the statement and proof of Lemma 3.36, r^{2r} should be replaced with O(r)^{3r/2}, and d^{2d} replaced with O(d)^{3d/2}.  in the last two equations in the display, replace mes(R^d/Gamma) and 2d.d! both by O(d)^{3d/2}.
• P. 144: In the last pragraph of the proof of Theorem 3.38, the superfluous ) should be deleted.
  
• P. 145: In Theorem 3.40, replace “If d >= 2 and P is not proper” with “If d >= 2, P is not proper, and Z is torsion-free”.  In the proof, “We induce on d” should be “We induct on d”.  On the last line, “unless Z is torsion-free” should be deleted.
• P. 147: In the second paragraph, "M_1,...,M_{d-1}" should be "M := (M_1,...,M_{d-1})".
• P. 155: In Q 4.1.7, E_{xi in H^perp} should be sum_{xi in H^perp}.
  
• P. 158: In the proof of Lemma 4.10, xi/a should be xi a.   Between the second-to-last and third-to-last lines of the long display in that proof, insert the lines “= \P_F(A)^3 - |F|^{-1/2} \|f\|_{L^2(F)}^2” and “\geq \P_F(A)^3 - |F|^{-1/2} [ \E_{a \in A} \| 1_{a \cdot A} \|_{L^2(F)} ]^2”.
• P. 162: In the third line of Corollary 4.15, a_1,..,a_k should belong to A rather than F, and 2^{1-k} should be 2^{-k}(1 + |F|^{-1}).  (Remark: the order estimate only gives the qualitative result kA=F for |F| sufficiently large, but the small values of |F| can be done by hand.)  In the last word of the last sentence, F should be A.
• P. 164, in Q. 4.3.10, "A.2A = A" should be "A.2A=2A".
• P. 171, in Q 4.4.6, sqrt(2) should be 1/4.
• P. 177: in penultimate display, (1-eps^2) should be (1-eps)^2.  "If take" should be "Taking".
• P. 182, middle: "S' \in S_theta" should be "S' \subset S_theta". In the last display, the upper limit of the integral should be infinite.
  
• P. 183: In the last four displayed equations, all >- signs should be <=.  In (4.38), an (A) is missing from the RHS.
• P. 215, middle, "closed unit interval" should be "closed unit ball".
• P. 216, middle: "some origin a=x/2" should be "we see that $A \cap H_+$ contains a set F symmetric around some origin a=x/2".
  
• P. 218, fourth line: eps/K|A| should be \frac{eps}{K} |A|.
• P. 229: In the proof of Theorem 5.30, Corollary 2.23 should be Lemma 5.26.
  
• P. 231, first line of proof of Theorem 5.33: "|A| \leq 100 K^2" should be "|A| \geq 100 K^2".
  
• P. 263: In the statement of Corollary 6.20, the K^4 in the denominator in the last line should be K^5.
• P. 264: In the proof of Corollary 6.20, 16 sqrt(2) K should be 16 sqrt(2) K^2 (two occurrences), and K^4 should be K^5 (one occurrence).  In the proof of Theorem 2.29, K^4 should be K^5 (four occurrences).
• P. 294: In statement of Proposition 7.21, add "Furthermore, each of the w_i lies in the set {v_1,...,v_n}".
  
• P. 333: In Lemma 9.3, F_p should be F.
  
• P. 341: In Question 9.2.6, V should be Delta_n.
• P. 342: In Theorem 9.17, r_i should be |R_i|.  In the last sentence of the proof of Theorem 9.16, "B" should be "B := S_i".
  
• P. 344: In proof of Corollary 9.19, "the multiplicatively invertible" should be "the set of multiplicatively invertible".
• P. 352: In the proof of Lemma 9.31: all occurrences of g_1 should be a_1, etc.
  
• P. 370: Conjecture 10.6 should be attributed just to Erdos, and not to Erdos-Turan (in particular, reference [99] should be deleted; I am not sure what the earliest reference for this conjecture is).  Similarly for exercise 10.0.6 on page 372.
  
• P. 370: In Theorem 10.5, the phrase "... with |Z| coprime to (k-1)!" should be added at the end of the first sentence.
• P. 375: In the second and third displays, Lambda should be Lambda_3.  In the line afterwards, (4.3) should be (4.2).
  
• P. 378: Corollary 10.10 should be Proposition 10.10, and similarly in page 380.
• P. 379: in the third line of the proof of Theorem 10.12, 3/n should be 3|Z|/n.
• P. 381: In the second line, 4 should be 8.  (In the next two displays, the 4 is correct.)  In the third line, add "and x_0 in Z" just before ", such that".  In the fourth and fifth displays, E_Z(g) should be E_{Z'}(g).
• P. 382: in the paragraph after statement of Theorem 10.20: replace"delta and M" with just "M".
• P. 383: in the third display, the first = sign should be <=, while \sup_{\xi \in V^\perp \ 0} |\hat \nu(\xi)| should be \sup_{\xi \in V^\perp} |\hat \nu(\xi) - {\bf I}(\xi = 0)|.
• P. 384: in Lemma 10.22, eta needs to be between 0 and 1.
• P. 385: in third display, xi'-xi should be xi-xi'.  In the next display, the sum eta + I(xi-xi'=0) should be enclosed in parentheses.
  
• P. 384: In the first line, Theorem 10.12 should be Proposition 10.12; in the fourth line, Corollary 10.10 should be Proposition 10.10.
• P. 388, fifth line of proof of Lemma 10.25: insert ", since" after "On the other hand".  In the third-to-last line, "the sum" should be "the summand".
• P. 389: in top line, "f is bounded" should be "f is bounded by 1".  In third line, P should be P'.  In Exercise 10.3.1, add the hint "(You may want to first establish the weaker but easier bound r_3(P) = o(|P|).)"
  
• P. 503: Reference [352] should be redirected to [380].
  

Thanks to Mei-Chu Chang, Moubariz Garaev, Peter Komjath, Tom Koornwinder, and Sam van Gool for corrections.

Nonlinear dispersive equations: local and global analysis
Terence Tao
CBMS regional conference series in mathematics, July 2006
Softcover, 373 pages.  ISBN-10: 0-8218-4143-2, ISBN-13: 978-0-8218-4143-3

These lecture notes try (perhaps ambitiously) to introduce the reader to techniques in analyzing solutions to nonlinear wave, Schrodinger, and KdV equations, in as self-contained a manner as possible.  It is a six-chapter book; the first three chapters and an appendix can be found here.  It is based on these lectures.

Corrections to the printed version:

• Page 4: In the first paragraph, (6.4) should be 6.4.
• Page 54: In the “Conversely” portion of Exercise 2.2, one must assume the Lorenz gauge condition \partial^\alpha A_\alpha = 0.
• Page 55: For the Schrodinger equation, the phase velocity is half the group velocity rather than twice the group velocity (i.e. v/2 instead of 2v).
• Page 85: At and before (2.40), 4 pi should be 8 pi.
• Page 109: “defocusing, absent, or focusing” should be “focusing, absent, or defocusing”.
• Page 112: After (3.5), Z^d should be R^d.
• Page 135: In Proposition 3.15, T does not depend on k.
• Page 136: “1 < p < 1+4/n” should be “2 < p < 2 + 4/d”.  “q<r” should be “r<q” (two occurrences), and “p/q > 1/q’” should be “p/q < 1/q’”.
• Page 138: In (3.28), the H^1 norm should be on R^d, not on I x R^d.
• Page 139: In the second to last display in the proof of Proposition 3.19, the exponent 5/2p should be 5(p-1)/2.
• Page 140: In Figure 5, H^1 should be \dot H^1.
• Page 146: In the proof of Proposition 3.23, Proposition 3.23 should be Proposition 3.22.
• Page 150: “subcritical” -> “sub-critical”
• Page 151: H^1 should be H^1_x.
• Page 152: In exercise 3.35, the first appearance of “defocusing” should be omitted.
• Page 154: mu=-1 should be mu=+1.
• Page 156: After (3.37), Delta^2 a should be –Delta^2 a.  In (3.38), an integration in dt is missing. 
• Page 159: In (3.45), an integration in dx is missing.
• Page 166: L^q_x should be L^q_{t,x} (similarly on (3.51) in page 167).
• Page 167: W^{1,10/3} should be W^{1,10/3}_x.
• Page 174: In the first paragraph, (3.55) should be (3.56)
• Page 175: In the proof of Proposition 3.35, eps^2 should be eps^2/2 (two occurrences).
• Page 193: In (4.7), -5u^4 should be +5u^4.
• Page 208: Superfluous ) parenthesis on (4.18) and on the preceding equation, as well as the display two equations down.
• Page 220: In (4.13), L^2_t L^infty_x should be L^2_x L^infty_t.  In (4.14), L^4_t L^infty_x should be L^4_x L^infty_t.
• Page 339, second display: ++ should be +.  In final display on RHS, N^s M^{-s} should be N^{2s} M^{-2s}, and L^2 should be H^s.
• Page 340, equation (A.20): N^{-2k} should be N^{-k}. 
• Page 341, last display in proof of Lemma A.9: The L^2 norm on the LHS should be squared.
• Page 343, Exercise A.8: In the endpoint Sobolev inequality, both instances of the exponent d should be replaced by d/(d-1).  (Also, d needs to be strictly greater than 1.)
• Page 344, Exercise A.18: The hypothesis that u is spherically symmetric is missing.
• Page 348: Before (B.3): “a positive and finite” should be “positive and finite” 

Many thanks to Soonsik Kwon, Timothy Nguyen, Tristan Roy and Shuanglin Shao for corrections!

Honeycombs, hives, and puzzles; or, Hermitian sums, GL_n(C) tensor products, and Schubert calculus on Grassmannians
Allen Knutson, Terence Tao
American Mathematical Society
In preparation

This is a self-contained introduction to the combinatorial tools of honeycombs, hives, and puzzles, and their applications to linear algebra, representation theory, and algebraic geometry. This book will not be finished until 2008 at the earliest.