Advice is what we ask for when we already know the answer but wish we didn’t.(Erica Jong)

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
- High school level
- 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).

- 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.

- 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.

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?

More advice:

- John Baez’s page on career advice.
- Po Bronson’s article on the relative importance of innate intelligence versus effort.
- 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“.
- 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.
- AMS advice page for new PhDs
- 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.

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2 September, 2019 at 12:04 pm

ahartelHi,

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

AnonymousI 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

AnonymousIf 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.

2 September, 2019 at 8:02 pm

ahartelThank you

6 April, 2020 at 5:15 am

Gianmarco Brocchihttps://terrytao.wordpress.com/career-advice/on-the-importance-of-partial-progress/

7 September, 2019 at 12:19 pm

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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

JacquesDear Pr. Tao,

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?

Thank you.

29 April, 2020 at 8:33 am

portonConsider 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

Bo BerndtssonDear Jaques,

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.

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

ShobaThanks 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

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5 November, 2020 at 6:21 am

AnonymousMaryam (Iran mathematician) was very lucky to take the advice from here

11 December, 2020 at 6:42 am

aaronschumacherI 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

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7 January, 2021 at 2:06 am

Victor PortonDear Terrence. Career advice is important, but your post misses the global effect of research careers. Here is what affect the career issues of

oneresearcher 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

AnonymousTwo famous examples of academy-rejected top mathematicians are Galois and Ramanujan.

31 January, 2021 at 8:04 am

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9 February, 2021 at 2:45 pm

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17 April, 2021 at 1:46 am

MahdiDo 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 PhysicistStudy the math GRE with me.

18 April, 2021 at 8:43 am

AnonymousThe 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

Jacob WakemI 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 SilvermanRenyi’s comment might be apropos here.

30 October, 2021 at 5:57 pm

Jacob WakemSome brains simply cannot handle dopaminergic substances. It gives an illness called psychosis.

30 October, 2021 at 5:55 pm

Jacob WakemDear 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.

Example:

55555…

+22222….

27777…..

Sincerely,

Jacob Wakem

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14 January, 2022 at 8:37 am

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30 January, 2022 at 10:16 am

Jacob WakemHow 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 PhysicistI 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

AndrewJas, 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 PhysicistThank 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.

20 February, 2022 at 8:03 am

Jas, the PhysicistI’ve visited this page so many times, but maybe this time the advice will stick.

6 May, 2022 at 6:15 pm

Ujjwal KumarHi 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

AnonymousOf 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

AnonymousYou might remember this: After one stops asking advices from others, one is going to find out his own path. And love it.

21 May, 2022 at 12:50 pm

Jacob WakemThe 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

30 June, 2022 at 3:36 pm

AnonymousYou should have an article about stimulants.