Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Subscribe to feed In these pages are the latest information (including sample chapters and errata) for all the various books that I have been an author of: T. Tao, Solving mathematical problems: a personal perspective , Deakin University Press, 1992. T. Tao, Three regularity results in harmonic analysis , in “Topics in analysis and applications: selected theses”, R. Coifman, ed. World Scientific Publishing, 2000. T. Tao, Analysis I , Hindustan Book Agency, 2006. T. Tao, Analysis II , Hindustan Book Agency, 2006. T. Tao, Solving mathematical problems , second edition, Oxford University Press, 2006. T. Tao, V. Vu, Additive Combinatorics , Cambridge University Press, 2006. T. Tao, Nonlinear dispersive equations: local and global analysis, CBMS regional series in mathematics, 2006. T. Tao, Structure and Randomness: pages from year one of a mathematical blog , American Mathematical Society, 2008. T. Tao, Poincaré’s legacies: pages from year two of a mathematical blog , American Mathematical Society, 2009. (Includes graduate texts on ergodic theory and on the proof of the Poincaré conjecture.) T. Tao, An epsilon of room , Vol I., Graduate Studies in Mathematics, 117, American Mathematical Society, 2010; and An epsilon of room, Vol II., American Mathematical Society, 2010. (Includes a graduate text on real analysis.) T. Tao, An introduction to measure theory , Graduate Studies in Mathematics, 126, American Mathematical Society, 2011. T. Tao, Topics in random matrix theory , American Mathematical Society, 2012. T. Tao, Higher order Fourier analysis , Graduate Studies in Mathematics, 142, American Mathematical Society, 2012. T. Tao, Compactness and contradiction , American Mathematical Society, 2013. T. Tao, Hilbert’s fifth problem and related topics , American Mathematical Society. 2014. T. Tao, Spending symmetry , in preparation. T. Tao, Expansion in finite simple groups of Lie type , Graduate Studies in Mathematics Vol. 164, American Mathematical Society, 2015. T. Klowden, T. Tao, Climbing the cosmic distance ladder , in preparation. T. Tao, Six Math Essentials , Quanta Books, to appear, Nov 2026. I will unfortunately not be able to respond to any requests for full-length copies of these books, beyond the material already linked to in these pages. I have also been involved with the following books, though not as a primary author: The Princeton Companion to Mathematics , edited by Tim Gowers, J. Barrow-Green, I. Leader. Princeton University Press, 2008. Mathematicians: an outer view of the inner world , Mariana Cook, Princeton University Press, 2009. Komplexität und Universalität , Terence Tao and Jochem Berlemann, Die weltweit besten mathematischen Artikel im 21. Jahrhundert , Band 2, 2014. (A translated and illustrated version of this talk of mine .) 2016 MATRIX Annals , David R. Wood (Editor-in-Chief), Jan de Gier. Cheryl E. Praeger, Terence Tao, Springer, 2018. (Similarly for subsequent editions of the MATRIX Annals.) Do Not Erase: Mathematicians and Their Chalkboards , Jessica Wynne, Princeton University Press, 2021. Recent Comments A Severe Misalignment of AI in Mathematics "Deep theorems were scarce and difficult and so became an effective mechanism to identify deep thought. AI has broken this system." 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Polymath 13 – a success! Non-transitive Dice over Gowers’s Blog Comments feed for this article 29 September, 2015 at 2:56 am Hello,terrence professor.this is my first time to take a comment in your blog.I feel so excited as if you were in front of me。you are my idol.i love math.my dream is to be a best mathematician.but i think it`s difficulty .i want to understand the article in your blog.how could i do? 3 November, 2015 at 10:21 am In “Spending Symmetry”, p.11 you use the older-style terms “direct limit” and “inverse limit”, for which as you know the newer-style terms are, respectively (and confusingly!), “colimit” and “limit”. A suggestion: maybe indicate both (pairs of) alternatives. It might be a bit awkward, but this is a very common point of confusion. 5 November, 2015 at 6:41 pm nicholasgrabill Besides those listed, would you recommend any other books as nice mathematical reads? 17 February, 2016 at 2:40 pm Dear Professor Tao, In chapter 2 of Analysis I, the object n++ is never defined. Assuming it is a function, it is not stated until lemma 2.2.10 that every positive natural number has a unique pre-image. However, before that, when considering Definition 2.2.1 (definition of addition), the question does come to mind whether a preimage (any, not necessarily unique) can always be obtained from n++. The questions I have are: 1. Does definition of addition (2.1.1) assume that given a number not equal to zero, one can always work out the (unique) “predecessor” to it? 2. In attempting the proof of lemma 2.2.10 (Let a be a positive number. Then there exists exactly one natural number b such that b++ = a), how do I perform induction on a when, by statement of the axiom of induction, the property should be true for 0 but clearly here is true for all a except zero. I have always taken the first element available ( in this case 1) when doing induction. However, here I am trying to work from first principles, hence this question. 3. is “n++” a function? If so, it seems to have been implicitly defined through its characterisation by axioms of natural numbers. Is that a valid way of defining functions? Many Thanks for your time, 17 February, 2016 at 7:14 pm The definition of addition is recursive, and relies on Proposition 2.1.16, which in turn uses pretty much all of the Peano axioms. Strictly speaking, one does not directly use Lemma 2.2.10 in the proof of Proposition 2.1.16, but the two statements are certainly very consistent with each other (as you say, if there were a non-zero number with no predecessor, then it should not be possible to uniquely specify a value to that number from a recursive definition, although formalising this intuition rigorously requires a bit of work.) For your second question, you can perform induction on a statement such as “Either is zero, or there exists exactly one number such that .” For the third question, strictly speaking is an operation rather than a function, but it can be made into a function once one has a little bit of set theory available: see Example 3.3.2. 4 September, 2025 at 10:38 pm Pardon thread resurrection, but the link shows that operations are a type of function. Can some explain what I missed? Thanks 5 September, 2025 at 7:36 am I was drawing a (rather subtle and technical) distinction between the external notion of a function in the ambient metamathematics (e.g., the metamathematics required to talk about a model of Peano arithmetic), and the internal notion of a function in a theory that itself has a notion of sets (and functions, etc.), such as ZFC. In Chapter 2 of this text, where Peano arithmetic is discussed, there is no internal notion of a set or a function yet (this is introduced in Chapter 3), so operations such as ++ are only functions in the external sense, not the internal one. However, once enough set theory is introduced internally in the model, one can also view these operations as internal functions as well. This distinction is however mostly of relevance when doing model theory (particularly when considering “nonstandard” models of various theories, such as nonstandard arithmetic), and can be safely ignored by the “working mathematician”. If you are interested in precise logical foundations for the material covered in this book, you can work through the Lean companion for the text , which formalizes all of this material in the proof assistant language Lean. This distinction is however mostly of relevance when doing model theory (particularly when considering “nonstandard” models of various theories, such as nonstandard arithmetic), and can be safely ignored by the “working mathematician”. If you are interested in precise logical foundations for the material covered in this book, you can work through the Lean companion for the text , which formalizes all of this material in the proof assistant language Lean. 23 July, 2016 at 2:24 am Dear Professor Tao, I dont know if you know the beautiful blue Danube walz composed by Johann Strauß. The spark for the music was provided to him by a dream containing the words ” Viana be glad oh oh why why”. So dreams often contain some hidden dormant breakthrough. Stepping out of the mist into the dawn of a new beauty. Now my dream : 1+1= 2 or 1. 1+2= 3 or 1 or 2 1+3= 4 or 1 or 2 or 3. 2+2= 4 or 1 or 2 or 3. 2+3= 5 or 1 or 2 or 3 or 4. 5+7= 12 or 1 or 2 or………………..or 11. It has to do with groups and circles and the fact that any chess piece can be replaced by a musical instrument and any chess game can thus be turned into a musical composition of some sort, and waves combining up to form larger waves, or breaking up into many individual smaller waves. Thanks for a great website and your time ( youtube lectures etc…) 5 September, 2016 at 6:02 pm Dear Terence, when do you plan to AMS publish your multiplicative analytic number theory notes? By the way I feel they are your most impressive lecture notes, displaying the broadest array of skills and originality in their combination, do you see some other set of notes you are more proud of? (I suppose you will include related posts like that on the Montgomery uncertainty principle.) One more question: do you feel you could easily rework out all the results you have presented at least in lecture notes? I mean given the title of the notes, give the main results with proofs? Which would give you the hardest time? Congratulations and thanks for all the work. 2 October, 2016 at 3:49 am Repository Alpha For those who are not aware Analysis I & II have been published in their third edition at Springer, https://www.springer.com/gb/book/9789811017896 @Terry What are all the differences between this edition and the second one? Thanks! 15 September, 2024 at 7:00 am For those who are interested in particular on the theory of orthogonal curvilinear coordinates (Lamé was) a new book has been published that introduces a new perspective through matrix operations. The title: “Orthogonal curvilinear coordinates [and more] for engineers and physical scientists” http://orthogonal-curvilinear-coordinates-book-ts.com/ 18 January, 2017 at 4:38 am Respected Tao sir I have a general doubt on whether harmonic series of natural numbers have any direct asymptotical formula with the p nth prime. 5 April, 2017 at 8:55 pm I find it amazing that you have written all these books considering you are a Fields Medalist and do so much researce.there are different types of books this is great work so far. 14 December, 2017 at 12:59 pm I am looking for a integral sol for damped wave equation in R^2 with full Cauchy data. Thanks 3 January, 2018 at 4:07 pm I tried to purchase your Analysis I and II (published by springer). Unfortunately, they say it is out of print. It appears it was never printed by Springer, since I cannot find it on Amazon either. This is unfortunate because I want to make it a required text for our analysis class! Of course, students have access to the electronic book through the library, but sometimes it is nice to have a physical copy to scheme through. I found scheming to be very useful when I was a student, as it led to reading sections that were not necessarily covered in class (granted I’m old-fashioned). Best wishes, Masoud University of Queensland 3 January, 2018 at 11:38 pm The texts are published by the Hindustan Book Agency. 21 March, 2018 at 4:40 pm Do you know if the books (Analysis I & II) will be printed in the future by Springer ? If so, when ? 19 February, 2024 at 11:39 pm On trouve sur le net GRATUITEMENT les 2 pdf d’Analyse en anglais 10 January, 2018 at 7:00 am Dear terry sir, i would like you to see the doubt that i have stated above because i am interested in knowing any approximation to the nth prime related to or dependent on the harmonic number. And any book about prime number 12 January, 2018 at 3:40 am stuff list – incarnation […] https://terrytao.wordpress.com/books/ […] 14 May, 2018 at 8:39 pm I have been using your Analysis text book this semester. It has been generally great. One shortcoming I have noticed is with the multivariable section. I have found that the students need a good amount of discussions about continuity and limits. This is because some subtle issues arise; e.g., it could happen that a limit does not exist, even if it exists if one approaches on any line, or one can have a discontinuous function which is continuous in both variables x and y (if one fixes the other variable), etc. It would be great if a chapter is devoted to discussing multivariable limits and continuity before diving into derivatives. Some examples such as xy/(x^2+y^2) or x^2y/(x^4+y^2) as (x,y)–>(0,0) would help students. Thanks, Masoud University of Queensland 15 May, 2018 at 6:38 pm This is discussed in Section 2.2 of Analysis II (see in particular Exercises 2.2.9-2.2.11, as well as Example 1.2.7 of Analysis I). 20 May, 2018 at 3:46 pm Thanks for the reply. I think a couple of more exercises discussing the R^2 situation may be helpful; e.g. is a function which is continuous in every direction continuous? is a function continuous along every path to (0,0) actually continuous at (0,0)? …. On another note, the version of the inverse function theorem in the book discusses differentiability of f^{-1} only at f(x_0). In applications one would like to know that f^{-1} is differentiable on the neighbourhood V. How should one think about this discrepancy? [Fair enough. I’ve added a few exercises to explore these statements in the errata for the book. -T] 8 June, 2018 at 10:46 pm Create a blog for your audience today – a step-by-step guide – Constantinuous […] of the research mathematician, Professor Terence Tao (UCLA) is very inspiring; he wrote several research monographs and co-authored books (integrating past blogs and notes) and solved open problems through that blog […] 21 August, 2018 at 6:01 pm CARLOS ALBERTO CARDENAS CANO CORDIAL SALUDO, MI NOMBRE ES CARLOS ALBERTO CARDENAS Y QUISIERA PEDIRLE EL GRAN FAVOR DE QUE POR FAVOR RECOMENDARA LIBROS DE CALCULO DE PRIMERA CALIDAD. PERDONE POR MI IGNORANCIA NO SOLO EN MATEMÁTICAS SINO TAMBIÉN EN SU IDIOMA, PERO CREO QUE UNA PERSONA COMO USTED PODRÍA RECOMENDAR LIBROS DE PRIMERA CALIDAD Y RIGUROSIDAD PARA EL PUBLICO QUE TIENE LA AMBICIÓN DE TENER UNA BIBLOGRAFIA QUE SEA LA MEJOR PARA SU ESTUDIO Y PARA EL BENEFICIO DE LA HUMANIDAD: CREO QUE HACE FALTA RESEÑAS DE LA MATEMÁTICOS PROFESIONALES COMO USTED ACERCA DE LOS MEJORES TEXTOS PRODUCIDOS EN EL MUNDO COMO LOS SUYOS. MI ADMIRACION Y RESPETO. MUCHAS GRACIAS 9 November, 2018 at 9:25 am El problema es que, por cada materia, existen tantos libros que se hace dificil nombrar o enumerar los mas buenos y completos ya que siempre hay varios. Aunque es cierto que esos libros ‘clasicos’ siempre existen. Tan solo hay que buscar en Internet recomendaciones en foros, google, Math Stackexchange, Quora, Reddit… Le recomiendo hacer busquedas en su buscador de Internet tales como ”best books for…(asignatura)” . Otra cosa importante es intentar no redactar su mensaje todo en mayúsculas. Y también intentar aprender (como mínimo) a leer/entender, y escribir el inglés ya que es el idioma de la ciencia por el cual nos entendemos todos. El inglés científico realmento no es muy dificil ya que constantemente se repiten las mismas expresiones (sobre todo en textos matemáticos). Un saludo Carlos Alberto y mucha suerte y ánimo en sus estudios ! 9 November, 2018 at 8:18 am Dear Professor Tao, Is it possible to contact you by email and ask you some questions about Maths? Yours sincerely, Vitalii Frolov 9 November, 2018 at 9:36 am I don’t know what kind of questions you want to solve but… unless you have a very important doubt or something related with reasearch/academia/his books… it would be better to leave Professor Tao alone. We shouldn’t make him to lose his valuous time. That’s the best for Mathematics/Science and the Mankind. 2 July, 2019 at 7:46 am The link for “Mathematicians: an outer view of the inner world” returns a 404 error: https://press.princeton.edu/titles/8860.html [Link updated, thanks – T.] 31 May, 2020 at 9:56 am Dear Prof. Tao, I wanted to thank you for the amazing books you have published. I have worked through many of them (Analyses, MT, Epsilons) very thoroughly, painfully solving each exercise – without these books I would have never learned to appreciate the true beauty of mathematics. Your exposition is brilliant and motivated, one cannot return to other manuscrpits after working with yours. I personally know professors who decided to switch to your books, and I have read dozen of positive feedbacks from mathematicians on social media regarding your unique way of presenting undergraduate and graduate level material. I know how extremely limited your resources are. Nevertheless I will continue to hope that you will publish your other lecture notes (e.g. Linear Algebra) as books too someday. A lot of students are using these materials, and it would make a great impact. 31 May, 2020 at 10:06 am Esp. books in “classic” subjects are very helpful, since they are taught at almost any university in the world. Like linear algebra, undergraduate differential equations, probability, topology. Some of them already exist as online lecture notes on your blog (probability theory for instance), but having them as books would be just amazing for students. But again, I understand completely that you do not have time to spend on writing books on these “trivial” topics. 12 September, 2020 at 1:22 am Dear Terry, considering the significant number of revisions made since the publication of the Analysis books in 2014, are you planning a fourth edition? I contacted the Hindustan Book Agency, and they told me that the publication of the fourth edition is dependent on you sending them the revisions. I am asking since I plan to order a big number of copies, and I would rather order a revised version. Thanks a lot! [I currently have no short-term plans for a fourth edition (mostly due to lack of time on my part), but may do so in the future, particularly if the list of errata stabilizes. -T] 26 September, 2020 at 9:33 pm it used to be a comment here. i just wanted to reply to that but couldn’t find it. i forgot posters name. i just wanna say thanks to her/him, that website is really good for listing used textbooks. it gives a free dedicated channel. thank you again for all of you. 28 August, 2025 at 6:43 am could you leave the link to the website 1 September, 2021 at 3:30 am (Yes I bought both of them) 12 January, 2022 at 5:45 pm Consideraciones Preliminares – Cosas Matemáticas […] El primer experimento que haremos es acerca de los hipervínculos. Dejaré las instrucciones: primero hay que escribir algo, luego lo seleccionamos con el cursos y damos click izquierdo. Seleccionamos «enlace» y agregamos eso. Ejemplo: Aquí está el blog de Terence Tao. […] 10 March, 2022 at 12:23 am Jas, the Physicist I feel like I’m going to be reading this blog for the rest of my life. 3 March, 2023 at 1:23 am Can I learn real analysis on my own? - StudentMovementUSA […] Real Analysis by Terence Tao […] 31 May, 2023 at 11:40 am edinsonbohorquez130 I have a question Professor Terence Tao. In which language is it better to learn mathematics? In English, Spanish, Hindi or some other language? 9 July, 2026 at 12:51 pm Dear Prof. Tao, The link for “ Six Math Essentials ” doesn’t seem to be working. [Fixed, thanks – T.] 10 July, 2026 at 9:49 am Dear Professor, the link for “Six Math Essentials” still does not work. [As per the notice at the top of this page, this page will be maintained going forward at https://teorth.github.io/tao-web/#books Leave a comment Cancel reply To enter in LaTeX in comments, use $latex <Your LaTeX code> $ (without the < and > signs, of course; in fact, these signs should be avoided as they can cause formatting errors). 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