This site is currently hosting

- updates on my mathematical research;
- expository articles (such as my articles for the Princeton Companion to Mathematics, or for the tricks wiki);
- discussion of open problems;
- talks that I have given or attended (such as the Distinguished Lectures Series at UCLA);
- my advice on mathematical careers and mathematical writing;
- information about my books and applets;
- my lecture notes on ergodic theory, on the Poincaré conjecture, on random matrices, on graduate real analysis (245A, 245B and 245C) and introductory graduate probability (275A), on Hilbert’s fifth problem, on expansion in finite simple groups of Lie type, on higher order Fourier analysis, and on analytic number theory;
- and various other topics, usually related to mathematics.

While most of the posts are aimed at those with a graduate maths background, I will also occasionally have a number of non-technical posts aimed at a lay mathematical audience. My selection of topics is guided by my own personal taste; I do not take requests for specific topics to post about on this blog.

I welcome comments from people with all kinds of mathematical backgrounds and levels of expertise; my only requests are that the discussions are kept constructive, polite, and at least tangentially relevant to the topic at hand. Comments which are spam, self-promoting, off-topic, or otherwise not fulfilling the above requests will be summarily deleted; repeated offenders in this regard may be subject to blocking. In particular, comments devoted primarily to promoting one’s own research are subject to deletion. Also, comments which essentially duplicate previous comments may also be deleted, or used to replace the previous comment, as appropriate. Finally, comments whose sole purpose is to solicit an answer to a homework problem are discouraged, and will be deleted if they are unlikely to lead to any discussion of wider interest. (However, questions *inspired* by a homework problem, for instance inquiring as to further connections between two mathematical topics connected by such a problem, or questions centred on a very specific technical point in the solution of that problem, are welcome.) Of course, I will not be able to personally respond to all the comments made on this blog.

I have enabled the ability to rate comments on this blog. However, this rating system is unregulated and should not be taken as any sort of official evaluation of one’s comments. In the interest of constructive criticism, negative ratings should be used sparingly. To discuss the ratings system, please visit this thread.

Any discussion, feedback, questions or suggestions not related to one of these topics can be placed as a comment to this “About” page, or at my open thread. Comments about formatting and presentation can be made at this page.

Terence Tao

[*Update*, Mar 31 2007: Opened this page to comments.]

— Some technical remarks —

WordPress has the ability to insert LaTeX math code (e.g. ) into both posts and comments. The format for this is “$`latex `*[Your LaTeX code]*$” (without typing the square brackets). See this announcement for details. Note that LaTeX macros and environments are *not* supported, similarly, double dollar signs $$ do *not* create LaTeX displays (one can use \displaystyle to get an approximation of these displays, though.) Also, line breaks are not allowed within a LaTeX code.

There used to be a number of quirks with the WordPress LaTeX plugin, but they have now largely been fixed. If you find any problems, please report them at this page.

WordPress also supports a certain amount of HTML. As a consequence, be careful with using the signs in a comment, they may be misinterpreted as HTML tags! You can use < and > instead.

In case a comment really gets mangled up by formatting errors, you can contact me and I can try to manually correct it.

I have heard that it is possible to configure wordpress so that comments can be previewed; if anyone has any specific knowledge on how to implement that feature, I would appreciate knowing about it. [*Removed*, Apr 8 2007, in response to comments.]

If a comment does not immediately appear after you submit it, it may have been accidentally flagged as spam (this in particular can happen for a post with an excessive number of links). In that case, please contact me and I will de-flag it.

I do not have PDF copies of my posts. However, the “print preview” feature in your browser should convert the post to a format which is suitable for conversion to PDF, with the sidebar and header removed. Also, at the end of some years, I convert many of my blog posts into a book format; see this page.

I have occasionally been asked for the formatting I use for my own posts. I use the Tarski theme with a modified CSS, in order to do things such as boxed theorems. (To use the CSS, one needs to purchase a CSS upgrade.) I also use Luca Trevisan’s LaTeX to WordPress converter to write the more mathematically intensive posts.

— Copyright etc. —

Readers are welcome to copy, link to, quote from, or translate reasonable portions of the content of this blog (e.g. a single article) into other media, though for items longer than one or two paragraphs, I would appreciate it if a reference or citation to the URL that the content originates from is provided. If you wish to copy a significantly larger fraction of the content (e.g. an entire series of articles), please contact me about it first.

## 233 comments

Comments feed for this article

31 January, 2017 at 9:17 am

Open thread for mathematicians on the immigration executive order | What's new[…] About […]

26 February, 2017 at 12:04 am

IB_Math AnalysisHi Terry and other users:

“I welcome comments from people with all kinds of mathematical backgrounds and levels of expertise; my only requests are that the discussions are kept constructive, polite, and at least tangentially relevant to the topic at hand. Comments which are spam, self-promoting, off-topic, or otherwise not fulfilling the above requests will be summarily deleted; repeated offenders in this regard may be subject to blocking. In particular, comments devoted primarily to promoting one’s own research are subject to deletion. Also, comments which essentially duplicate previous comments may also be deleted, or used to replace the previous comment, as appropriate. Finally, comments whose sole purpose is to solicit an answer to a homework problem are discouraged, and will be deleted if they are unlikely to lead to any discussion of wider interest. (However, questions inspired by a homework problem, for instance inquiring as to further connections between two mathematical topics connected by such a problem, or questions centred on a very specific technical point in the solution of that problem, are welcome.) Of course, I will not be able to personally respond to all the comments made on this blog.”

Acknowledged.

IB

27 September, 2017 at 4:55 am

Maths studentDear Prof. Tao,

during the remainder of the holidays I will embark on writing an open source content management software similar to wordpress, but with several features differentiating it from the same.

1) The HTML code will be highly optimized so as to allow slow (but even fast) computers to render the page significantly more quickly

2) In order to insure optimized HTML code, the edit mode will not be a WYSIWYG editor, but rather a JavaScript-powered construction mode, where you have building blocks for each HTML element, with at most some hybrid structure for situations where the HTML structure is very clear

3) There will be made use of code indentation so that the code of the website will be readable

4) The code will be compliant to used standards, so that there will be browser compatibility

5) Special care will be taken so that the website enlarges properly and the screen is properly used

6) Since I intend to use the software, it will from the beginning be designed to support mathematical formulae in LaTeX form

7) I’m thinking about an advanced maths captcha system

Since you’ve run a mathematics blog for a considerable amount of time, I’d be very grateful if you could state features that you felt were indispensible, and post a list of the wordpress add-ons that you use.

I would even be willing to try and implement features that are somewhat non-trivial, but even though I wouldn’t require it, it would probably not take too much time to have maths lectures (perhaps even at a graduate level) videotaped and published, so that they can be watched by mathematicians all over the globe; for instance, Prof. Steve Butler had his calculus course uploaded to YouTube (https://www.youtube.com/watch?v=YoVgNE4lKSk&list=PL1BE3027EF549C7D1). I suspect that very many people might be interested in such a thing, and the benefits seem to me to outweigh the costs.

23 October, 2017 at 12:06 am

Maths studentAlright, if you wouldn’t like to answer that’s fine, I just would find it useful for the elicitation of requirements (I’m working (vaguely) according to the waterfall model in accordance with my own aesthetic preferences).

23 October, 2017 at 12:11 am

Maths studentOf course I did move on to more advanced modelling for time reasons.

16 November, 2017 at 11:16 am

IB_Mathematical_Pinion (not pinyin)A search facility on the this blog must surely be a debatable topic?

https://codex.wordpress.org/Creating_a_Search_Page

It’s the first hit in a google search, this recommendation is coming from a realization that one sometimes needs to refer to a long list of comments and pull out a quote, not necessarily one of Terry’s either, often other users make highly useful contributions to the blog. Not that one is trying to be overly unselfish, three blog posts in one day? That’s pretty selfish.

21 November, 2017 at 12:21 pm

The MinionI found the search facility in between the most recent comments sections and the articles by others sections, it’s good, must have been there for the blog’s entire existence, a lesson in patience.

16 November, 2017 at 1:08 pm

IB_Mathematical_Minion (not pinyin)Just related to my words about selfish posting on the blog, I think it’s getting pretty clear that you just have to be yourself on this site, Terry has done a sensational job of organizing it and it is certainly absolutely a terrific concept.

This post relates to the about section because it relates to site use, site users definitely need to feel positive about using the site and free to develop their own senses for the subject, I see the WordPress blog by Terry as his own brilliant way of meeting the demand for his teaching and his work has been efficient and exemplar in meeting that demand and producing a site that is achieving a subtle and significant contribution to the site user’s mathematical experience.

20 November, 2017 at 8:12 am

andyTo whom it may concern,

I was scrolling through the page at “terrytao.wordpress.com/tag/boolean-algebra” and happened to click on a link in the side bar. Under the “Software” heading, the link for “Papers” is currently “http://www.mekentosj.com/papers/”. This is broken and when “papers/” is removed (leaving just http://www.mekentosj.com/), it redirects to “https://www.readcube.com/papers/”.

[Link now deleted – T.]12 December, 2017 at 6:07 am

Anonymousdear jenious prof terence”

is age a factor too? for learning maths ,

i have been learning maths since two years and i am 17 years old ,

before that i didn’t know what is(1/2)

i am about to complete single variable calculus now,

and you are an inspiration for me (Y)

22 January, 2018 at 3:01 pm

CryptoMan(tm)Prof Tao, Would be interesting to hear your thoughts on crypto currencies and some of the maths behind it? e.g page 21 of https://ethereum.github.io/yellowpaper/paper.pdf

Appendix: E.1. zkSNARK Related Precompiled Contracts. We choose two numbers, both of which are prime.

(240) p ≡ 21888242871839275222246405745257275088696311157297823662689037894645226208583

etc. Does it pass the Tao sniff test?

9 March, 2018 at 3:17 am

AksharaHi,

For a layman, can you prescribe a list of books/ basic approach to be able to acquire graduate level mathematics on their own?

Do you think that is possible?

I’d love to learn mathematics (among other subjects) for myself, and I’m looking for a way to approach this.

Thanks,

A

9 March, 2018 at 2:33 pm

zgtzsjIt highly depends on your current math level. For me, I would choose a online course and try to follow it to see whether I am interested in it and handle it.

18 March, 2018 at 7:37 am

Siddharth NayakDear prof Tao,

I am a high school student. I have many mathematical concepts like calculus l and ll, trigonometry, complex numbers, etc.

My question is ” how can I understand mathematical equations, i.e, what the equations are saying and how to build the equations logically to get the solution”.

13 May, 2018 at 8:55 am

AnonymousDear Professor Tao,

During the course of your mathematics career, have their been any problems that captured your attention more than others? Or did they all hold equal weight for you?

Sincerely,

Jonathan Hasan

13 May, 2018 at 8:56 am

AnonymousEqual weight in terms of your interest in them.

13 May, 2018 at 11:45 am

EdmondHello professor Tao ,

I am planning to participate in the next IMO . I would like to improve my problem solving skills and learn more advanced topics so i can be prepared. What are the books that you can recommend me and are mathematical thinking books an important part for preparation?

16 May, 2018 at 10:16 am

Math Blog and how to write math equations using LaTeX $latex…$ | Adonis Diaries[…] stated on his About page, he uses Luca Trevisan’s LaTeX to WordPress converter to write his more mathematically […]

7 June, 2018 at 4:43 am

Jonathan JoestarI am high school student student and now preparing for entering university.Is it ok to research math in graduated school when I major other science area?

Also if you do not mind, please tell me what you major in university.

7 June, 2018 at 5:49 am

Jonathan JoestarSorry about greeting you.

Dear Pro.Tao

5 September, 2018 at 11:42 pm

Anthony MudhokaI am Anthony,a math lover and your greatest fan from Kenya.I am preparing for my A levels.I have always been captivated by patterns in numbers especially primes.There is a question I have always been thinking about but I don’t get a quite satisfactory answer when I search it up.What are the implications if an explicit formula for the nth prime was to be found today?How would it improve our current understanding of math?Would it be accessible to the public since it renders internet security useless?Is to going to be a big feat for humanity or will it be treated just like any other formula?I am looking forward for your answer.

4 December, 2018 at 12:52 pm

Candace FrenchHi Terry,

Not sure of how to get ahold of you, except on here.

I am looking to order one of your books. Can you recommend which one would be best, for someone who is really into prime numbers?

It is a Christmas present and is it possible to have you autograph it?

Thanks so much!

Candace

11 January, 2019 at 11:02 am

Artur (@ArturTweeting)Dear Professor Tao,

Now that many mathematicians and math bloggers are on Twitter (including Professor Gowers who has recently started tweeting as @wtgowers), do you consider joining Twitter as well?

Many thanks in advance for the response.

29 May, 2019 at 3:44 pm

Sebastian HellmanHello Mr Tao,

Thank you for your great work and efforts during you career.

I work on the subject how to accelerate the development of new mathematics and mathematical physics.

If you where given the task to go as fast as possible from 7th grade level of math to 9th grade level how would you do?

In what time do you think or know it has been done?

The same question from 9th grade to 2nd year undergraduate level/start of the second year at the University-level?

If you just wanted to be a good student at that level as fast as possible from the 9th grade level how would you do?

It would be terrific if you found the time for a reply.

Thank you from Sweden and from Sebastian and my friend Marsha.

25 June, 2019 at 2:23 am

Hung TranHello Terence, I’m looking for an expert on the Riemann Hypothesis. I’d like to get in touch, if you’d like.

25 June, 2019 at 12:25 pm

AnonymousHi Prof. Tran,

It may be harder to get an expert to review your purported proof of RH than to prove RH. Good luck!

I downloaded a copy of your proof… Thanks! And I am certain RH is true!

26 June, 2019 at 7:57 am

AnonymousNature is a grand optimizer, and therefore, one and only one critical line, Re(z) = 1/2, (the Riemann Hypothesis), is enough to define all primes and all nontrivial zeros of the Riemann Zeta Function. Hmm. Do you agree?

17 August, 2019 at 9:39 am

A bit of googology – Julia Wolffe's Notes[…] I learned, from Terry Tao’s About page, that WordPress quite straightforwardly supports LaTeX math code. I’ve also been wanting to […]

12 September, 2019 at 9:32 pm

JohnDear Professor Terence Tao,

I am a massive fan of yours. I am an average electrical engineering student hoping to improve his math abilities by getting into visualization techniques, and I was wondering, what do you see when you close your eyes and visualize mathematics? Do you see a tree-like structure, do you see lots of colors (like Richard Feynmann seeing lots of symbols enshrouded in colors), or do you see a memory palace? Or do you see something completely different? And how do you actively review these memories so you don’t forget them?

~With Warmest Regards,

John

12 September, 2019 at 10:50 pm

AnswerHere is your answer:

15 September, 2019 at 1:54 pm

Bob CopelandI read your conditions for comment but I thought I’d take a chance anyway. I’ve accidentally discovered how Tesla’s(9) theory of numbers can classify all the numbers from zero to infinity into groups, strings, fields, quantum time lines & rotation. I’m not highly educated from your viewpoint & abilities, but sometimes the fickle finger of fate flicks something mathematically profound into the most stupid places to the most stupidly qualified. If you want to take a chance or flyer just contact me or anyone else for that matter at bcope16@hotmail.com.

14 October, 2019 at 6:01 pm

Ali SadaHi Prof. Tao,

Is the algorithm below connected to your work on the Collatz Conjuncture? It was tested for 1.2 * 10^6.

1. pick an integer n>0

2. If n is even, divide by 2. If n is odd, find the least perfect square m2 greater then n and add m2+n.

3. Repeat step 2. with either n/2 or n+m2

The conjecture is this algorithm will always terminate at a 1 or an 11. Why do 1 and 11 appear? Is there anything significant about the number of “termination points” (here, just 1 and 11, so 2 termination points.)?

Generalization: The algorithm seems to terminate for all n when replacing the least perfect square greater than n with the greatest perfect square less than n. It also seems to terminate when square is replaced by any power.

Best,

Ali Sada

15 October, 2019 at 3:23 am

AnonymousThe least perfect square greater than is at most which (by a simple probabilistic model) implies a negative drift for the logarithm of the orbits, thereby indicating the possibility of proving that almost all orbits are bounded.

25 November, 2019 at 6:12 pm

Sharath MathewDear professor Tao, I need your input on this math problem.

I was wondering if you could help me as everyone I seem to bring this mathematical problem to can’t seem to understand it or what I’m talking about. So here it is: with regards to linear congruential generator and middle square algorithm I am missing four components to the equation. What are they? The equation is simple. If you google LCG you will be able to see which equation I’m talking about. I need a seed. A multiplier. A increment. As well as a modulus. The equation is as follow: Xn+1= [(x*a)+c]modm

In the above equation I don’t know how to type subset so it is x subset n plus 1 (not x multiplied by n plus one).

From these pools of numbers three have a single number pulled from a pseudorandom number generator. Without knowing the seed (starting number), multiplier(A), increment(C), or the modulus(M) how will I know what number is to be pulled out of the blank spots 1,5 & 6? Is it possible to figure out these missing variables using the given numbers; 28,021, 48,124, & 64,869? So can you find out what is the next number in this sequence? It’s supposed to be unpredictable but was made from a deterministic computer. Meaning a number such as 64,869 was a result of the number previous to that number being used in that same equation which was 48,124. If so, please tell me how you did it and lay out the steps you took.

The numbers given to me are as follows:

1—-00,000-16,339

2—-16,340-32,678 28,021

3—-32,679-49,017 48,124

4—-49,018-65,356 64,869

5—-65,357-81,695

6—-81,696-98,036

Kindest regards,

Sharath 281-995-5091(c)

Remyzale@yahoo.com

8 April, 2020 at 12:27 pm

Steven HsuDear Professor Tao,

I learnt about your existence from a Youtube video that popped up in my feed at random.

Compared to you, I am light years away in terms of my math skills.

I’ve never been good at math because I always had a hard time understanding the questions asked – especially the written ones. It’s just hard to visualize – especially when they started throwing in symbols on top of verbal questions.

My math level as of right now is probably no higher than grade 10-11 since after highschool, I went straight into the Fine Arts in university.

I always thought perhaps my brain was wired differently since birth.

Math has always been something I looked at with much curiosity and interest, but it was just very alien to me. I did suffer from ADHD and depression during my earlier years in education so this could be one of the reasons I had a hard time focusing in my math classes as well. Sorry for going off topic here.

I’m not exactly young – currently 35 – but I do feel I still have maybe some brain left to casually learn math on my own time. Just something I feel I want to prove to myself that I could at least understand the nature of mathematics beyond the elementary arithmetics.

One thing I’ve always done reasonably well in math, however, was in geometry. I suppose the direct visuals helped. My poorest results were direct equations – in particular long equations. Perhaps poor memory…

Professor Tao, do you have any kind of books that you could recommend to me for situations such as this?

Thank you for your time!

– Steven H.

14 April, 2020 at 11:05 pm

DanielDear Steven,

I can relate to your journey.

I am 36 this year. We can perhaps collaborate on something to get your mathematics discovery journey going.

You can reach me at

dctogoaustralia2011@gmail.com

Daniel

1 June, 2020 at 3:56 pm

Anonymousassuming is it possible to see Corona virus with our eyes ( without the help of a microscope ) how can you kill the virus if it is impossible for the virus to survive around the sun surface.

use the knowledge of physics and mathematics to prove your result.

10 June, 2020 at 11:04 pm

{RCRISTO - Tecnologia e Informação}Dear Tao,

in my studies of mathematics, I find empty space as the source of everything we can imagine and represent, especially in mathematics. What is your opinion in this regard. Are all spaces born empty?

Thank you for your attention.

8 August, 2020 at 9:01 pm

aaronUse boinc. Designed by Berkeley. Design a project to run the numbers for you. Thousands of people will download it and all their computers will be pumping out results to you. That’s what they designed it for. It was originally designed just for SETI but now its used for climate change projects, health research, etc. Check it out.

Boinc.berkeley.edu

Mass computing for science.

14 August, 2020 at 4:18 am

Island of Blue-Eyed People | Logos con carne[…] I started following Terry Tao’s blog, and his “Selected Articles” list includes a post about this puzzle. I read it, and […]

1 October, 2020 at 1:25 pm

AnonymousThere are lots of places suddenly showing “formula does not parse”. The display of some equations is distorted. Is there anyone who knows what is going on?

11 February, 2021 at 11:14 am

LaTeX – extra examples and some bugs | Machine Intelligence ?[…] with a discussion of peculiar issues. Some other also interesting remarks can be found in the About page of Terence Tao’s blog What’s […]

16 February, 2021 at 12:53 pm

ElliotMy theory for Math is that the most part of advanced concepts are lies, and you need to discover by yourself how the things works.

16 February, 2021 at 1:56 pm

Thomas EninbergIt’s really, the most people don’t know how the first humans reached the division count, and it’s very interesting :D

7 March, 2021 at 6:40 am

DTHi Prof. Tao, I found that your Problem Solving book has been uploaded as a PDF at

Click to access solving-mathematical-problems-terence-tao.pdf

Are you ok with people using this?

6 June, 2021 at 5:39 pm

Charles CollupyI built a Prime Number & Factor Machine

How can I Help?

https://www.desmos.com/calculator/x1p87euik1

Cheers

15 June, 2021 at 4:10 pm

Benjamin Mirad GuriniHello Terence,

I like the Collatz Conjecture too. In my attempt at finding some sort of pattern, I have been (thankfully) introduced to many other amazing conjectures.

Most of these revolve around primes.

Seeing that primes have so many amazing properties, I can’t help but wonder; Why are primes made? What are the underlying “glitches” or properties of integers, such that these primes, and their subsequent patterns I made?

I understand, that this is essentially what all proofs are working towards. But do we have any general idea why primes are so weird? And could they potentially be linked to the other number types, just in a way we haven’t figured out?

18 September, 2021 at 7:57 pm

Viki Esther ChangHi Terence, just wondering if you would like to come onto my podcast for an Interview on Education, 20mins, Regards, Viki Esther, Singapore-France

https://youtube.com/channel/UCqS436JuhIepBK8lF7VibEA

19 September, 2021 at 7:00 pm

Anonymousif i have a nice solution for your 8-4-2-1 problem or whatever collatz problem were can i send it to?! also copyright only me dont want my shit stolen! to give u a hint it really is simple u just use higher dimesional thinking in like using earths rotation for tracking time meters or lightspeed! just define c max, meter and 1 second and u will be able to find your problem with physics! very paradox! so what i can offer is using (24x60x60) – (23 x 56 x 4) = 81248

prefection reality

to help u 8 is like 180degrees eight infinity 1-2-4-8 is your collatz prob cos the rotation difference between you and your primne universe is off! ending in increase of enthropy immediate shutdown or lockdown and complete chaos systeme failure !

19 September, 2021 at 7:01 pm

greg matterhornif i have a nice solution for your 8-4-2-1 problem or whatever collatz problem were can i send it to?! also copyright only me dont want my shit stolen! to give u a hint it really is simple u just use higher dimesional thinking in like using earths rotation for tracking time meters or lightspeed! just define c max, meter and 1 second and u will be able to find your problem with physics! very paradox! so what i can offer is using (24x60x60) – (23 x 56 x 4) = 81248

prefection reality

to help u 8 is like 180degrees eight infinity 1-2-4-8 is your collatz prob cos the rotation difference between you and your primne universe is off! ending in increase of enthropy immediate shutdown or lockdown and complete chaos systeme failure !

26 February, 2022 at 8:56 am

The International Congress of Mathematicians (ICM 2022) will now be fully virtual. How best to utilise this format? | What's new[…] About […]

23 April, 2022 at 2:37 am

caii have a question all the time.that’s how a mathematician learn and do math？and how hight level they reach？for example，seven

Theorems of continuity of real numbers，can

they easily prove them pairwise？are these kept in their mind？whether they think mathematical analysis and linear algebra and so on are too easy？how could i learn math well？could you answer me？thank you！

19 May, 2022 at 2:31 pm

Juan MarchionattoProf. Tao,

I am an amateur mathematician who worked on the Collatz conjecture for the last 15 years. I think I have a demonstration, but I have not been able to publish it on arXiv or any other site to have it somehow registered before exposing it for comments, because, not being related to a university, I haven’t been able to get an endorsement.

Would you endorse me at arXiv so I can publish my solution there?

Of course, I would send it to you before, at your request.

Thank you in advance.