Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Subscribe to feed Here is my collection of various pieces of advice on academic career issues in mathematics, roughly arranged by the stage of career at which the advice is most pertinent (though of course some of the advice pertains to multiple stages). Disclaimer : The advice here is very generic in nature; I don’t pretend to have any sort of “silver bullet” that will solve all career issues. You will of course need to evaluate many factors, contexts, and needs specific to your own situation, as well as employing a healthy dose of common sense, before making any important career decisions. I would in particular recommend discussing such decisions with your advisor if you have one, as he or she will be familiar with your situation and will likely be able to provide pertinent advice. Also, it should be clear that most of this advice is targeted towards academic careers in mathematics; of course, there are many other career options available besides this, but I have no particularly informed advice to offer for such alternatives. Primary school level Advice on gifted education . Advice on gifted education . High school level Advice on mathematics competitions . What universities should one apply to ? Advice on mathematics competitions . What universities should one apply to ? Undergraduate level How can one become better at solving mathematical problems ? Note that there is more to maths than grades and exams and methods ; there is also more to maths than rigour and proofs . It is also important to value partial progress, as a crucial stepping stone to a complete solution of a problem. Don’t base career decisions on glamour or fame . But you should study at different places . Does one have to be a genius to succeed at maths ? My commencement address to Harvey Mudd graduates in 2022 ( video ). How can one become better at solving mathematical problems ? Note that there is more to maths than grades and exams and methods ; there is also more to maths than rigour and proofs . It is also important to value partial progress, as a crucial stepping stone to a complete solution of a problem. Don’t base career decisions on glamour or fame . But you should study at different places . Does one have to be a genius to succeed at maths ? My commencement address to Harvey Mudd graduates in 2022 ( video ). Graduate level It is important to work hard , and work professionally . But it is also important to enjoy your work . Think ahead to understand the way forward; ask yourself dumb questions to understand the way before. Attend talks and conferences, even those not directly related to your own work . Talk to your advisor , but also take the initiative . Don’t prematurely obsess on a single “big problem” or “big theory” . Write down what you’ve done , and make your work available . In this regard, I have some advice on how to write and submit papers . “ A close call: how a near failure propelled me to succeed “, T. Tao, Notices of the American Mathematical Society, August 2020. Originally contributed to “ Living proof: stories of resilience along the mathematical journey “, American Mathematical Society, 2019. Eds: Allison Henrich, Emille Lawrence, Matthew Pons, David Taylor. It is important to work hard , and work professionally . But it is also important to enjoy your work . Think ahead to understand the way forward; ask yourself dumb questions to understand the way before. Attend talks and conferences, even those not directly related to your own work . Talk to your advisor , but also take the initiative . Don’t prematurely obsess on a single “big problem” or “big theory” . Write down what you’ve done , and make your work available . In this regard, I have some advice on how to write and submit papers . “ A close call: how a near failure propelled me to succeed “, T. Tao, Notices of the American Mathematical Society, August 2020. Originally contributed to “ Living proof: stories of resilience along the mathematical journey “, American Mathematical Society, 2019. Eds: Allison Henrich, Emille Lawrence, Matthew Pons, David Taylor. Postdoctoral level Learn and relearn your field , but don’t be afraid to learn things outside your field . Learn the limitations of your tools , but also learn the power of other mathematician’s tools . In particular, you should continually aim just beyond your current range . In your research, be both flexible and patient . You should definitely travel and present your research if given the opportunity. But be considerate of your audience ; talks are not the same as papers . Be sceptical of your own work , and don’t be afraid to use the wastebasket . Learn and relearn your field , but don’t be afraid to learn things outside your field . Learn the limitations of your tools , but also learn the power of other mathematician’s tools . In particular, you should continually aim just beyond your current range . In your research, be both flexible and patient . You should definitely travel and present your research if given the opportunity. But be considerate of your audience ; talks are not the same as papers . Be sceptical of your own work , and don’t be afraid to use the wastebasket . I am also (slowly) in the process of gathering my thoughts on time management from the perspective of a research mathematician. Here are some general thoughts on this topic. Batch low-intensity tasks together to take advantage of economies of scale and to reduce distraction. What are some useful, but little-known, features of the tools used by professional mathematicians? John Baez’s page on career advice . Sara Billey’s “ How to get a Ph.D. in a timely fashion “. Po Bronson’s article on the relative importance of innate intelligence versus effort. Catherine Cannizzo’s “ Postdoc applications “. Fan Chung’s advice for graduate students . Lance Fortnow’s “ Graduate Student Guide “. Oded Goldreich’s “ On our duties as scientists “. Richard Hamming’s “ A stroke of genius: striving for greatness in all you do “. Matt Might’s “ Illustrated guide to a Ph.D. “ Gian-Carlo Rota’s “ Ten lessons I wish I had been taught ”. J. Michael Steele’s “ Advice for Graduate Students in Statistics .” Ian Stewart’s “ Letters to a Young Mathematician “. Ravi Vakil’s “ For potential students “. Remy van Dobben de Bruyn’s “ Timeline of postdoc applications “. Lauren Williams’ “ Finding a postdoctoral position in mathematics “. The Princeton Companion to Mathematics ‘ section on advice to younger mathematicians , with contributions by Sir Michael Atiyah , Béla Bollobás , Alain Connes , Dusa McDuff , and Peter Sarnak . Early career articles in the Notices of the AMS AMS graduate student blog The Mathematics Stack Exchange has a number of questions and answers on career development (and one can ask further questions that have not already been posed on that site). MathOverflow similarly has questions and answers on careers . Finally, the Academia Stack Exchange has a large number of questions and answers on all academic matters, including career issues. My page on views on AI also contains some relevant advice on these topics. 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Logic reading seminar (8) The sciences (1) The Polymath Blog Polymath News and AI Polymath projects 2021 A sort of Polymath on a famous MathOverflow problem Ten Years of Polymath Updates and Pictures Polymath proposal: finding simpler unit distance graphs of chromatic number 5 A new polymath proposal (related to the Riemann Hypothesis) over Tao’s blog Spontaneous Polymath 14 – A success! Polymath 13 – a success! Non-transitive Dice over Gowers’s Blog Comments feed for this article 2 September, 2019 at 12:04 pm do you happen to have a copy of your Google+ post “value partial progress, as a crucial stepping stone to a complete solution” that you could re-post on this blog? I just wanted to look it up but since Google+ closed down I can’t read it anymore. Regards, Andreas 2 September, 2019 at 1:54 pm I happened to have saved the text before Google+ was taken down. It can be found here: https://pastebin.com/MkzSrdYm . 2 September, 2019 at 4:26 pm If Google+ can be closed down and you can’t read things on it anymore then that could happen to anything. I wonder if anyone is saving the text from this WordPress site. 17 February, 2026 at 11:13 am Exactly what the Internet Archive and Wayback Machine are for, isn’t it? 2 September, 2019 at 8:02 pm 6 April, 2020 at 5:15 am Gianmarco Brocchi On the importance of partial progress 7 September, 2019 at 12:19 pm پیشنهادهایی برای دانشجویان تحصیلات تکمیلی سیستم‌های پیچیده - سیتپـــــور […] تائو یک مجموعه نوشته خوب در مورد شروع تحصیلات تکمیلی داره که پیشنهاد می‌کنم […] 26 October, 2019 at 8:07 pm Física y Matemáticas: consejos profesionales – ns137 […] Career advice https://terrytao.wordpress.com/career-advice/ […] 29 April, 2020 at 8:28 am After reading you career advices I feel like I am not able to find the right problems on which to work on, even if I own PhD (currently a postdoc). Most of the problems I encountered or asked myself eventually fell into two distinct categories: trivial problems (easy adaptation of known results, not even connecting two ideas from two distinct papers) and too hard problems (on which I have no way to start, or where the problems rapidly starts to be unsolvable). This is a serious problem as it gives the feeling that I am constantly not aiming above my range. It is almost like what I finally produce is too simple, and what I can not do is to difficult. Finding a good math problem to work is indeed a really difficult matter. Over the three years of my phD program, I followed the same path several times: I was given a problem that was too hard, then spent six months almost doing nothing (but still working 6/8 hours a day, always trying the same ideas that did not work) and ended up proving results that I knew how to prove at the start of the project. This is very frustrating, and it has a tendency to happen every time, even after the completion of the phD. I feel like I am not a real mathematician, in the sense that I do not improve the difficulty of the problems I am able to solve. To be fair, I identified a number of problematic situations: (i) I am not able to ask problems outside my very narrow field (despite the fact that I attend a lot of talks and conferences). Indeed I often have the impression that other fields are blocked. For example, I am working in a field very close to dispersive PDEs, but I am not able to find any problem on which to work on in dispersive PDE, a sort of problem that would extend my range and introduce me to this field (although I read courses on the matter, that does not help more than understanding line by line the papers of the field). It is almost like other fields seem “blocked” to me. (ii) I tend not to see doable problems when they exist, even in my very field of expertise. I had the feeling that some problems in my field were out of reach; but some mathematicians looked at the very same problem, used quite the same techniques as the one I knwo and solved the problem. (iii) I attended several conferences, some of which with the idea to discuss with people about their work and possibly launch work together: I read the papers in details, “learned” some of the techniques. But again, after reading the papers I had the (wrong) feeling that nothing more can be done. None of these attempts to work with other people has worked. (iv) I also know some people working in related fields and discussed various works with them, but yet, after 4 years of mathematics, not a single question to work on with other people, or mysels has emerged. Compared to what other mathematicians tell me, that is a strange situation. (v) I also feel like I need to rush and fear the non productive times, because I do not have a tenure position, so I need to prove that I am able to solve problems. But this has the drawback that I would not spend too much time on a problem without a serious lead, otherwise I would be afraid to not produce any paper by the end of the year. All of that shows that, even if I try a lot, I must not have the right approach on how to find problems to work on. I am also often told that many mathematicians work on problems that turn out to be too hard but manage to use what they proved to solve a part of the problem, or another problem resulting in a nice work (not oustanding, but nice). This seems like a very nice way of working, which never happened to me. Do you have any advice that would help me to find tractable interesting problems? 29 April, 2020 at 8:33 am Consider participation in Algebraic General Topology research https://mathematics21.org/algebraic-general-topology-and-math-synthesis/ – This is a breakthrough but very simple research: It is based on a set of simple axioms that were missed by decades by mathematicians. You could laugh if you see how much simple axiom sets were missed. Therefore, this is both easy and research productive. Yes, both easy and breakthrough. 3 June, 2020 at 1:24 am I think you describe very well what many or most mathematicians have felt at some point and of course I do not have any easy solution. But here is one thing that has helped me several times: Instead of learning a new technique and try to apply it to your problem, take a known result (that you like) and try to find your own proof of it. You may say that this is not very heroic but it can actually be quite useful, for yourself and others. At any rate it helps you to understand the theorem better, and if you find a nice proof it can be of interest to others as well, and it can even lead to generalizations of the theorem you started with. 20 January, 2024 at 1:03 pm Terence mentioned about asking dumb questions, these dumb questions maybe something you could just create from how the problem may possibly be or just imagined, from here now you can try to use what you know to answer the dumb questions, this may give you a pretty good chance of playing around with the problem and actually solve it. I guess just ask yourself a lot of interesting questions about the problem and it doesn’t necessarily lead to a solution but it makes everything better, it boosts the beauty of the problem. Sometimes you might get too stuck but that’s what I like because it boosts me to be too creative and think harder. 3 September, 2020 at 6:56 am The Last Stand: Custer, Sitting Bull, and the Battle of the Little Bighorn | Barbara Oakley […] given his youth. Tao is the winner of the 2006 Fields Medal and is a MacArthur Fellow. Here is his collection of various pieces of advice on academic career issues in […] 5 October, 2020 at 10:18 pm Thanks for sharing informative article. I was looking such kind of article. Keep sharing more articles like this.This article will help my sister who is currently studying at CGC Landran ( https://www.cgc.edu.in ) 21 October, 2020 at 8:52 am Adnan Khan is a qualified teacher and private tutor with 20 years of experience in mainstream education. science and maths tutor He saw the need for afterschool supplementary tutoring which would fill the gaps and surpass students learning to a much greater standard. This has been proven with his excellent success rate. 5 November, 2020 at 6:21 am Maryam (Iran mathematician) was very lucky to take the advice from here 11 December, 2020 at 6:42 am aaronschumacher I think the link for “Don’t be afraid to learn things outside your field” is currently broken because of an apostrophe. This link works for me: https://terrytao.wordpress.com/career-advice/dont-be-afraid-to-learn-things-outside-your-field/ [Corrected, thanks – T.] 23 December, 2020 at 7:45 pm Algumas idéias sobre como estudar Matemática – o Laplaciano […] (dados por alguém com muito mais experiência do que eu :)) podem ser encontrados na página Career advice do blog do brilhante matemático Terrence Tao (em […] 7 January, 2021 at 2:06 am Dear Terrence. Career advice is important, but your post misses the global effect of research careers. Here is what affect the career issues of one researcher may have on the global level: https://www.reddit.com/r/academia/comments/kpe3jm/how_modern_scientific_ethic_may_block_a_science/ 7 January, 2021 at 5:34 am Two famous examples of academy-rejected top mathematicians are Galois and Ramanujan. 31 January, 2021 at 8:04 am 43 Interesting Math Articles and Papers to Read in 2021 [Free PDFs] […] Advice (A Collection of Some very good advices) https://terrytao.wordpress.com/career-advice/Terence […] 9 February, 2021 at 2:45 pm Resources – Brian's Blog […] Anything from Terence Tao’s blog […] 17 April, 2021 at 1:46 am Do you have any career advice for someone who hasn’t done any maths for over 10 years and graduated with a MSc, but would just love to get back in maths and pursue a PhD? Thanks. 20 February, 2022 at 8:04 am Jas, the Physicist Study the math GRE with me. 1 May, 2024 at 4:00 am Jas, the Physicist What subjects of math do you enjoy? I’m past the math GRE now, but we can revisit. There are a lot of interesting integrals there. 18 April, 2021 at 8:43 am The advices here are, in my opinion truly helpful. But because the advices are aimed at mainly students or researchers who are going through formal mathematics education, I, just a high school student who has interest in Maths, had to find advices I could use which are scattered around those pages. If possible, please add some more advices for high school level, or if giving more advices than now to high school students is inapropriate, please explain why. Thank you. 1 June, 2021 at 7:32 am Essential Steps of Problem Solving in Mathematical Sciences […] Career Advice By Terrence Tao […] 28 October, 2021 at 9:13 am I think you should add to the burnout section. You don’t mention dopaminergic substances like large coffees or amphetamines. 29 October, 2021 at 10:48 am Robert Silverman Renyi’s comment might be apropos here. 30 October, 2021 at 5:57 pm Some brains simply cannot handle dopaminergic substances. It gives an illness called psychosis. 30 October, 2021 at 5:55 pm Dear Prof. Tao, My favorite system of infinite numbers (where system is left undefined) gives each digit a position (and also an order). Naturally, there are infinitely many digits for each number. You can line the numbers up to each other so you can add them up using carry addition. From this its easy to define infinite prime numbers. 55555… +22222…. 27777….. 1 August, 2024 at 11:32 am There are no prime numbers in this system. It does form a vector space though. 3 January, 2022 at 9:13 pm Buy latest unstitched 2Pcs Lawn Suits for Summer 2022, Designer 2 piece Embroidered Lawn Suits with Free Home Delivery all over Pakistan. 14 January, 2022 at 8:37 am This Week in CFD - Another Fine MeshAnother Fine Mesh […] some academic career advice from mathematician Terence Tao. Do not let the labels academia and mathematics keep you from […] 30 January, 2022 at 10:16 am How did you get your mathematics “discovered”? Of course you were semi-famous as an olympiad competitor, but how did you “hit it big”? 20 February, 2022 at 7:59 am Jas, the Physicist I left the University of Washington because I didn’t want to live in Seattle anymore. I was a sub-tier grad student who was depressed and lacking direction (not really). But I’d like to go back to grad school now. I really want to take the math subject GRE. Last time I sat the test, I filled in answers randomly. That was dumb, I should have just answered B the entire test, but I wasn’t think much about it. I’m going to try to get a good score this time. Not that anyone cares I guess, but because I want to prove to myself that I’m not afraid of entrance exams anymore. This will prepare me for quals/prelims, I hope. 30 January, 2023 at 3:01 pm Jas, the Physicist, I have heard from multiple sources that it’s very difficult to go back to graduate school to finish a PhD after leaving. I saw on your blog that your goal is to finish a mathematics PhD. I had several friends who left MIT before finishing their PhD, and none of them returned to finish the degree. I’m sure that many students have accomplished it, and I’m also not saying this to try to discourage you from re-applying. I’ve just heard that re-starting the process is a very challenging endeavor from both the academic and psychological perspectives. 3 February, 2023 at 3:13 am Jas, the Physicist Thank you Andrew for the comment I really appreciate it. The hardest part about Analysis for me is that I too heavily depend on R1 for visuals and since there are so many pathologies in higher dimensions I must correct this habit. I mean it’s a good heuristic but honestly it’s not realistic… hehe. 1 May, 2024 at 4:02 am Jas, the Physicist I’m back in graduate school and it’s much easier this time around. I reflected on why it didn’t work last time. 20 February, 2022 at 8:03 am Jas, the Physicist I’ve visited this page so many times, but maybe this time the advice will stick. 6 May, 2022 at 6:15 pm Hi Terence I am really confused in life about what career I choose , recently I had been amazed by the beauty of mathematics on internet and also watched a lot of your videos but after seeing to many views on internet I get demotivated because of all these videos related to children starting at age of 5-6 years competing in imo and several other Olympiads practicing for 3-4 years for IMO with natural talent , great IQ , discovered their passion early in life and getting accepted to good Universities like Harvard , Mit and Princeton etc. Really feels like jealous and thinking I wish I had do this or that . Do you think there is chance for 11th grade student to get a gold medal in imo and accepted in good Universities and become good mathematician or can become great mathematician without any of this and natural talent . (Sorry for my terrible English) 9 May, 2022 at 9:51 am Of course you can. Not every mathematician has what you described, and not every “child prodigies” became great mathematicians. Life is a long journey, and many things can go wrong, or detour the ideal path one wishes to have traveled. You may not win a medal in imo, but you can still become a mathematician. Most mathematicians are just average. They got a PhD. somewhere and got an academic career afterwards. 9 May, 2022 at 10:29 am You might remember this: After one stops asking advices from others, one is going to find out his own path. And love it. 1 August, 2024 at 11:26 am Most colleges are good colleges. One problem is many of them require survival skills to make it through (in addition to academic ability). Check out Jacob Wakem on researchgate.net for some bold, creative mathematics that doesn’t take a trained mind or 160 IQ to understand. I would strongly consider working on weekends, getting a driver’s license, etc. You don’t want to be a smart person who gets left behind. Your relationships also matter. Your mind will probably develop very quickly in college. It is these headspaces that are the real reason for doing mathematics in my opinion. 21 May, 2022 at 12:50 pm The continuum hypothesis can’t be resolved mathematically. Thus it is to be resolved philosophically. Then it is a philosophical statement. But no mathematical object has a merely philosophical existence. Thus there are no intermediate cardinals and the continuum hypothesis is true 1 August, 2024 at 11:17 am This comment isn’t Fields Medal worthy, but there is some good thinking condensed in it. 30 June, 2022 at 3:36 pm You should have an article about stimulants. 22 June, 2023 at 8:07 pm Dear Prof.Tao and other experts in this page, I’m current a phd student in medical science and my interest is the topology of brain network. To understand the topology, I move back to learn math. I’ve noticed that mathematical analysis is foundation which is noted by many answer about learning mathematics. However, I still confuse for it. Why mathematical analysis is important and what kinds of thing should be learn through it? Looking forward to your reply! Best Regards, Raphael 23 June, 2023 at 8:43 am First, learn English. It should be “I am still confused about it” It will take you a long time to learn the basics of analysis. It is better to collaborate with a mathematician with your research. 25 September, 2023 at 4:33 am hi，professor tao, i’m a 33 years old guy from china and i’m very interested in mathematics, is it too late for me to study math ? is there any areas i can work on? i do self studying, and now just finish my mathematical analysis course(using zorich’s book) 25 September, 2023 at 5:04 am I studied math analysis by Zorich book, too. Here is my research in which you can participate: https://math.portonvictor.org/algebraic-general-topology-and-math-synthesis/ 1 August, 2024 at 11:16 am One of my best classmates was a married 30 year old. It is possible you still have your gift. You should eventually feel it working in your head, proving you have the gift. 5 April, 2024 at 1:02 am Do you think Professor June Huh, who recently won the Fields Medal, is a genius? 25 May, 2024 at 1:04 pm 我的学术生涯小记 – Junfan Huang […] Terence Tao […] 7 July, 2024 at 9:09 pm Thanks for sharing such informative article. https://www.cgc.ac.in/ 8 September, 2024 at 6:42 am Ricardo Rodríguez Franco You solved the Collatz Conjecture but you haven’t realize. 26 September, 2024 at 2:45 am Nice piece of advice 7 November, 2024 at 3:36 am Still not spinning in his grave Speaking (or rather writing) of careers: I myself have opposed the serial rapist who may now become your president in every way I could. Unfortunately, my means were limited by my particular personal situation. Unfortunately, my means were limited by my particular personal situation. 12 January, 2025 at 3:58 am This is such a solid guide for anyone pursuing a math career! 💡 Maybe you could add some tips for those thinking about moving into industry or combining math with other fields? Stuff like how the problem-solving and analytical skills from math can fit into other careers would be super helpful. Also, some advice on balancing solo work vs. teamwork during grad school or postdoc would be cool. Another is information overload. I read this technique called Your topic, multiple stories by Brad . Worked for me. Great resource overall—thanks for sharing! 🙌 19 February, 2025 at 3:17 am That is one solid advice coming from a great such as yourself, Dr. Tao. Thank you. 26 April, 2025 at 8:07 am Dear Prof. Tao, The AMS advice page for new PhDs and AMS graduate student blog that you linked do not seem to work anymore. I could not find their current links myself. Could you update them? Thank you, Guanyuming He 29 June, 2025 at 1:45 am Thank you, Terence Tao, for inspiring us with your wisdom and dedication to mathematics! 4 July, 2025 at 10:44 am Dear Professor Tao, In your A Close Call: How a Near Failure Propelled Me to Succeed, you mentioned that the near failure propelled you to cut back on your gaming. Would it be possible that you point out some techniques to manage the desire to game effectively? Currently, I struggle to cut back on my gaming, too, which takes too much time from me. But all that I could have are temporary successes: I tried to uninstall the game a few times, and could hold for about one week each time. But during the period, my desire to play again increases, especially when I am tired or exhausted. Thus, not long after I uninstalled the game, I installed it back again to play, and then uninstalled again due to my disappointment on myself. The fact that you could successfully manage this is really admirable. I am hoping to learn how to break this harmful addiction from you, if you are willing to share. Thank you very much. 22 August, 2025 at 12:35 pm Remy van Dobben de Bruyn’s link is hopelessly broken (points to “ http://Remy van Dobben de Bruyn“) [Corrected, thanks – T.] 3 September, 2025 at 7:21 am it´s stilll not fixed…. i imagine something could be consistently mangling the target of the link [Fixed properly this time – T.] 14 October, 2025 at 8:36 am What is your opinion of closely following the arXiv? Is it good to read it everyday, or just check the papers journals or attend seminars regularly? 14 October, 2025 at 9:33 am The important thing is to stay informed of current developments in your chosen field; the precise mechanism in which one does so is of secondary importance, but generally having a diverse set of sources of information (including the three you listed, as well as conferences and talking to colleagues, and even some social media from reliable sources) is recommended. 10 November, 2025 at 5:10 pm What do you think about taking philosophy courses in undergrad or reading Shakespeare in high school? 4 February, 2026 at 10:19 am Lauren William’s “ Finding a postdoctoral position in mathematics “. This is a typo, her surname is “Williams”. [Corrected – T.] 2 April, 2026 at 6:16 pm With your distinguished accomplishments in the mathematical sciences, could you share what has enabled you to maintain your commitment and passion for mathematics throughout your career? 7 August, 2026 at 4:15 am Hello, I have found that in the Collatz conjecture, when the starting point is m=(2^n)*k+2^(n-1)-1, after 2n-3 steps it can reach a multiple of 4, denoted by x. Under certain conditions, the number of factors of 2 in x can be greater than n, and at the same time, this x is the highest peak in the iteration process. However, I do not know how to prove it. 16 August, 2026 at 10:15 am How would you describe your research workflow (especially with regards to exploring outside of your field), and has it changed substantially since the rapid improvement of LLMs at synthesising and summarising mathematics? As a secondary question: what fields do you believe are most/least likely to see the greatest acceleration in research due to AI? 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