Even fairly good students, when they have obtained the solution of the problem and written down neatly the argument, shut their books and look for something else. Doing so, they miss an important and instructive phase of the work. … A good teacher should understand and impress on his students the view that no problem whatever is completely exhausted.(George Pólya, “How to Solve It“)

One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution.

Learning never really stops in this business, even in your chosen specialty; for instance I am still learning surprising things about basic harmonic analysis, more than ten years after writing my thesis in the topic.

Just because you know a statement and proof of Fundamental Lemma X, you shouldn’t take that lemma for granted; instead, you should dig deeper until you *really* understand what the lemma is all about:

- Can you find alternate proofs?
- If you know two proofs of the lemma, do you know to what extent the proofs are equivalent? Do they generalise in different ways? What themes do the proofs have in common? What are the other relative strengths and weaknesses of the two proofs?
- Do you know why each of the hypotheses are necessary?
- What kind of generalizations are known/conjectured/heuristic?
- Are there weaker and simpler versions which can suffice for some applications?
- What are some model examples demonstrating that lemma in action?
- When is it a good idea to use the lemma, and when isn’t it?
- What kind of problems can it solve, and what kind of problems are beyond its ability to assist with?
- Are there analogues of that lemma in other areas of mathematics?
- Does the lemma fit into a wider paradigm or program?

It is particularly useful to lecture on your field, or write lecture notes or other expository material, even if it is just for your own personal use. You will eventually be able to internalise even very difficult results using efficient mental shorthand; this not only allows you to use these results effortlessly, and improve your own ability in the field, but also frees up mental space to learn even more material.

Another useful way to learn more about one’s field is to take a key paper in that field, and perform a citation search on that paper (i.e. search for other papers that cite the key paper). There are many tools for citation searches nowadays; for instance, MathSciNet offers this functionality, and even a general-purpose web search engine can often give useful “hits” that one might not have previously been aware of.

See also “ask yourself dumb questions“.

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7 March, 2008 at 10:06 am

Meeting on 7th March with Prof. Zhao «[…] Learn and relearn the field: learning the strength and weakness of tools, learning what else is going on in mathematics; learning how to solve problems rigorously. […]

14 June, 2008 at 11:32 am

这等牛人也在wordpress上写blog！ « Just For Fun[…] NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the […]

6 September, 2008 at 9:04 pm

AnonymousDear Prof. Tau,

First of all, I should thank you for your nice and useful hints and advices. Then, I would like to tell you a serious problem I’m facing with and ask you to help, if you don’t mind:

I’m a PhD student and my field is arithmetic algebraic geometry. As you know, the literature in this area is vast. So if I want to learn my field and go through all the details of all results and proofs, then I guess, I can never (i.e., in a reasonable time) work on my own thesis problem and produce anything. But, not learning and reading that way give me the feeling that I’m missing something and I’m not confident anymore.

I would really appreciate if you could give me some advice. Thank you in advance

7 September, 2008 at 8:48 pm

Matthew EmertonDear Anonymous,

I hope you won’t mind someone else providing you with some advice.

Most workers in arithmetic algebraic geometry (and not just students) suffer from the problem you describe to various degrees. The literature is indeed vast, and to read everything as a student, even everything that you might need in solving your particular thesis problem, is essentially impossible.

I would suggest the following: a good grounding in algebraic geometry is essential. Most students in algebraic geometry, of all flavours, go through the rite of passage known as “Hartshorne”: reading Hartshorne’s book, especially chapters 2 and 3, and solving vast numbers of exercises. It is more or less impossible, and in any case probably unwise, to avoid doing this. And once you have solved many/most of the Hartshorne problems, you should have some baseline confidence in algebraic geometry, scheme theory, and cohomology.

At the same time, there are other texts that it is good to look at because they emphasize certain functorial aspects of algebraic geometry more than Hartshorne, aspects which are particularly important in arithmetic algebraic geometry — e.g. Mumford’s red book. It is advisable to supplement your Hartshorne reading with such books.

Another standard text to read is Cornell-Silverman (and these days, depending on your precise direction of interest, Cornell-Silverman-Stevens — but this is more number-theoretic, while Cornell-Silverman is more geometric). This is not such a long book, and has a lot of information in it. Furthermore, since it is devoted to exposing Faltings’ proof of the Mordell and Tate conjectures, you get to see how all the geometric machinery is applied to solving a particular problem. As I’ll comment on more in a moment, this is crucial. (And I should also add, that there is no need to read this entire book — for example, below I will advocate skipping the chapter on Neron models, unless you really don’t want to.)

One thing that I would advise *not* doing, for most students, is reading large amounts of EGA and SGA. This takes a lot of time, and there is real danger of not getting anywhere substantive. In particular, it is safe, at least at the beginning of your career, to learn etale cohomology (say) as a black box. (Later, if it turns out that you need detailed information about how it is constructed, you can go back and learn them.)

What *is* worthwhile, is to get a good understanding of sheaf cohomology in the classical setting. (The beginning of Borel’s book on intersection homology, which ultimately is about perverse sheaves and so on, but which begins with background on constructible sheaves and Grothendieck’s six operations, is one place to do this.) The point is that *most* applications of etale cohomology use just the same sheaf theoretic formalism as one has in the classical setting (i.e. varieties over the complexes, with their complex topology), and the main technical theorems in the subject (proper base-change, smooth acyclicity, nearby and vanishing cycles) are precisely intended to show that etale cohomology, etale constructible sheaves, and Grothendieck’s six operations in the etale setting, behave exactly as they do in the classical setting. So if one has a good understanding of sheaves in the classical setting, you can be confident that your intuition there will carry over to the etale setting.

So one thing that is very much worth studying is Deligne’s first paper on the Weil conjecures. There you will see how he uses etale cohomology to prove a terrific theorem, and you will see that most of what he uses are properties that have perfect analogues (and are not so hard to establish) in the classical setting. So a good intuition for classical sheaf theory will let you understand a lot of the proof.

I could summarize this aspect of my advice as follows: spend time learning things that have a wide range of application (and thus take some advantage

of economies of scale): basic homological algebra and sheaf theory is one of these things — it underlies coherent sheaf cohomology (as in Hartshorne), etale cohomology and sheaf theory, perverse sheaves, D-modules, …, all of which are tools in arithmetic algebraic geometry. On the other hand, don’t spend lots of time learning technical details in a narrow direction until you are sure you will need them.

For the next aspect, I want to return to a point I made above: one way to learn an area is, rather than learning its technical details and foundations, is to learn how it can be applied to help solve problems. For example, p-adic Hodge theory is another tool which plays a big role in a lot of arithmetic algebriac geometry, and which has a technically formidable underpinning. But, just like etale cohomology, it has a very nice formalism which one can learn to use comfortably without having to know all the foundations and proofs.

Neron models of abelian varieties are similar: one almost never needs to use any facts about their construction (other than that they exist) when applying them. So it is safe to treat their existence as a black-box. (And if it turns out that you really need the proof for something, there is an article about it in Cornell-Silverman.) What is important is to understand how their existence can be used as a tool in other arguments. Because of my own mathematical background, the most natural place for me to point to is the literature on modular curves and modular forms due to Mazur, Ribet, and Wiles. In particular, the first couple of sections of Ribet’s famous Inventiones 100 article give a great example of how hundreds of pages of theory (including a lot of the theory of Neron models, and a quite a bit of SGA 7) can be summarized in ten or so pages of “working knowledge”.

If you ask other people, they will be able to give similar references for other topics that you might need, which summarize “everything you need to know” in a short number of pages, rather than the hundreds of pages of original sources.

Finally, what do you do to build your confidence, given that you’re skipping all these hundreds of pages?

For this, it’s good to remember that being a research mathematician is in any case not ulimately about reading and learning mathematics (although that plays a role), but about doing mathematics. So in some sense your confidence

as a research mathematician can (at least in principal) be somewhat orthogonal to the amount of foundational proofs you’ve assimilated.

What you need, rather (as I’ve already said above), is to understand how some important techniques can be applied to solve interesting problems.

One way to do this is by starting as soon as possible to look at the research literature.

Your advisor can suggest papers, and (depending on your precise interests) you can also choose “classics” of your own: Deligne’s first Weil conjectures paper, Ribet’s Inventiones 100 paper, Faltings’ paper in Cornell-Stevens, Serre’s paper in Duke 54 about his conjecures on modular forms and Galois representations, or any number of others. Try to find papers whose topic is appealing to you, which seem well-written, and which you feel you might have some shot of understanding something about (at least the statement of the main theorem) — but don’t expect to understand much of the technical heart of the paper at the beginning. Your goal is to get a feeling for how it is possible to marshal all the forces of the abstract theory to solve actual problems, by seeing someone else do it. It will take a lot of time and patience, and careful study, to do this — but in the end it should pay off.

One thing to pay attention to is the bibliography — it may lead you to other sources which explain necessary background material. Pursuing the necessary background by begining at the top and then working down, rather than trying to build everything from the ground up, is generally more efficient. (It is typically what working mathematicians do when they need to learn something new — begin with a paper of interest, and then go back into the foundational literature just enough to fill in those points they couldn’t understand from reading the paper itself.)

Another crucial way to build confidence (much more effective than learning new things!) is to solve problems yourself. You can begin with Hartshorne’s exercises, and any other exerices you can find scattered around. But at some point you will need more specialized problems to work on. You can ask your advisor to give you questions to solve. (And some advisors work this way in any case: rather than beginning all out with a thesis-level problem, they begin by having their students solve smaller, more manageable problems.)

But also, once you are looking at the research literature, you have an essentially endless supply of problems: just take any paper you are looking at, find a technical lemma whose hypotheses you can understand, and see if you can prove the lemma yourself. Try not to “cheat” by reading the given proof (but you may want to glance at what follows, just to check that the so-called lemma isn’t actually a five page argument). But, if you can’t do it yourself after a serious effort, you will be in a much better position to understand and appreciate the author’s argument — and whatever trick or technique they use

will be one that you probably will always remember in the future!

Doing this kind of exercise is one way that working mathematicians develop the skill of being able to glance over a paper in their field and then know essentially all the details of the paper. (A skill which I found extremely impressive when I was a student!)

And of course you can try to create and solve problems of your own. (Another skill which is important to develop.) Since I’ve already gone on much too long, I won’t say more about this here.

I hope this advice is of some use.

Regards,

Matthew

21 January, 2014 at 1:53 pm

GilAmazing. Thank you for such a thoughtful advice though I am not the OP.

8 September, 2008 at 9:29 am

AnonymousI really like Emerton’s advice. I wish I had this advice when I first studied arithmetic geometry. Nevertheless, it is still useful to me now even though I’m working in other fields. By the way, could you give the titles of Ribet’s Inventiones 100 paper and Serre’s paper in Duke 54?

9 September, 2008 at 8:35 am

Pete L. ClarkMatt Emerton’s advice is excellent. The only thing I might add is that Qing Liu’s recent book “Algebraic Geometry and Arithmetic Curves” looks like a legitimate alternative to Hartshorne for arithmetically minded students of algebraic geometry. However, it is true that the treatment of sheaves and cohomology in Hartshorne’s book is especially comprehensive (compared to Liu’s book and, for that matter, any other introductory text on the subject I know of, including also Mumford’s very fine Red Book), so I would recommend to a student to read selected parts of Chapters II and III of Hartshorne no matter what.

The references you asked for:

MR1047143 (91g:11066) Ribet, K. A. On modular representations of ${\rm Gal}(\overlineQ/Q)$ arising from modular forms. Invent. Math. 100 (1990), no. 2, 431–476. (Reviewer: Glenn Stevens) 11G18 (11F32 11F80 11S37)

MR0885783 (88g:11022)

Serre, Jean-Pierre(F-CDF)

Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overlineQ/Q)$. (French) [On modular representations of degree $2$ of ${\rm Gal}(\overlineQ,Q)$]

Duke Math. J. 54 (1987), no. 1, 179–230.

11F11 (11G05 14G15 14G25 14K15)

9 September, 2008 at 9:43 am

AnonymousPete L. Clark: Thank you very much for your comments and for listing the two papers.

A little comment regarding Hartshorne’s book: it’s a good book, but when I studied it for the first time, I struggled very badly with it. It was mostly my fault, but I think the book lacks crucial motivations in some parts. For example, the section on proper maps was especially tough for me; it never even mentions that properness is the analogue for compactness in topology. That may be obvious for someone who has studied complex manifolds or Riemann surfaces, but to a student who has not studied those, it’s hard. Another thing is that it’s also difficult to learn from Hartshorne how to think about schemes and their associated constructions such as fiber products. The functor of points concept is very useful for understanding schemes, but the book doesn’t discuss it. So in short, I’ve found that Hartshorne is good for someone who has had good preparation in geometry, but it’s hard otherwise.

9 September, 2008 at 9:43 am

Matthew EmertonJust to add one more suggestion for arithmetic geometry exercises:

Silverman’s two books on elliptic curves have a lot of terrific excercies, covering a wide range of number theory and geometry. (While the geometry is naturally focused to a large extent on curves, the second book has a chapter on elliptic surfaces, for example.)

These exercises vary quite a bit in difficulty, and are a good place to start for problems of a more arithmetic nature than those in Hartshorne.

And for reading: good survey articles can often give a huge increase in efficiency when

learning a new field. Even if the survey doesn’t give all the details, it will guide through

the more technical foundational literature, and save you flailing around with hundreds

of pages trying to sort out what’s what.

The standard places to find surveys are: ICM proceedings, and more generally,

other conference proceedings. In particular, the AMS’s “Proceedings of Symposia”

series has many great conference proceedings chock full of interesting survey

articles. In particular, every so often there is a big conference on algebraic geometry

in the U.S., and the proceedings appear in that series. Just writing from memory,

there is Arcata from `74, Bowdoin from `85, and the Seattle motives conference from

some time in the `90s. These have lots of surveys on all kinds of algebraic geometry,

including a lot of arithmetic geometry. There is also the great volume from the

early ’70s on the Hilbert problems (which includes Katz’s survey of Delignes’ proof

of the Riemann hypothesis over finite fields, and one of Langlands’ first papers on

Shimura varieties).

Reading surveys can’t substitute for reading more thorough treatments of a topic, but as well as giving a guide to these more thorough treatments, it also provides a way to learn quickly about different areas of mathematics, and so build up your general knowledge of mathematics. So I would recommend that as a grad student, it is good to supplement your detailed technical reading with well-written surveys on a range of interesting topics.

Regards,

Matthew

15 February, 2011 at 3:53 pm

anonymousThere is also Bourbaki seminar, where you will find surveys on recent advances in quite a number of areas.

9 September, 2008 at 10:01 am

Matthew EmertonDear Anonymous,

Regarding Hartshorne, I agree with you. For those trying to come to grip with algebraic geometry for the first time (and this typically means coming to grips with Hartshorne), I think there are various ways to try to make things smoother.

One is to read other texts like Mumford’s Red Book in conjunction with Hartshorne. It helps a lot with motivation for the underlying ideas. (Eisenbud and Harris is probably also good for this, although I don’t know it very well — I think it just post-dates my time as a grad student.)

The other thing to do is to make sure you have some understanding of classical topology and geometry. For example, there is a notion of proper map in topology: the preimage of a compact set should be compact. One can check that proper maps are precisely those that are universally closed (working in some reasonable category of topological spaces — let me not be to precise here, and thus avoid making a mistake!).

Similarly, one can check that a Hausdorff space is one whose diagonal map is a closed embedding.

In short, one can translate lots of familiar and simple topological and geometric notions into more categorical terms than those in which they are usually formulated. Having done this, the corresponding categorical notions in scheme theory become a lot more intuitive.

Unfortunately, I don’t know where this kind of comparison is made in the literature — I think it might be something that people who think about these sort of things rediscover for themselves at various points. Although it is not really a “trick”, it might be the kind

of thing that someone could write up for the “trick wiki”, or some companion wiki on

basic mathematical concepts, if such a thing is in the offing.

This is also the basis for my earlier recommendation about learning classical sheaf theory before learning the basics of etale sheaf theory. Etale sheaf theory is largely

about constucting a subsitute for classical constructible sheaf theory in the world of schemes. If you don’t know classical sheaf theory then you will be at a double disadvantage in trying to learn (or use) the etale theory — not only do you have to deal with the technical baggage; you don’t even know why you’re carrying all this baggage around!

One more comment on Hartshorne: many students, especially arithmetic geometry

students, tend to focus on Chapters 2 and 3 (which of course are the technical heart

of the book) and ignore Chapter 4 (on curves) and especially Chapter 5 (on surfaces).

But these final two chapters have lots of beautiful geometry in them, and reading them

can supply quite a bit of motivation to go back and do battle with Chapters 2 and 3.

And you also learn things that will help later in life. For example, understanding

models of curves over rings of integers (which, as schemes, are two-dimensional)

becomes much easier if you have a little feeling for the theory of algebraic surfaces!

Also, Mumford’s book “Lectures on curves on an algebraic surface” is absolutely

beautiful (as are all his books on geometry), and gives a lot of motivations (both intuitive

and technical) for all kinds of aspects of scheme theory (including Hilbert schemes,

Picard schemes, base-change theorems for coherent cohomology, the use of nilpotents

and deformation theory, … ). I highly recommend it.

Regards,

Matthew

9 September, 2008 at 11:46 am

AnonymousDear Matthew,

Thanks for more of your advice. As I said, I wish I had all of your advice when I first had the idea of learning algebraic geometry. It’s kind of late, but I still find many of your comments valuable now.

Regarding your suggestion on learning classical topological and geometrical notions in category theory terms, there is a set of lectures notes on AG by David Cox which does exactly that and is wonderful to read. One can find them here: http://www3.amherst.edu/~dacox/. It’s also important to include these ideas in a Wiki page so everyone knows where to find them.

Regards and Thanks

9 September, 2008 at 12:33 pm

Emmanuel KowalskiIt’s also worth noting that a lot of the algebraic and arithmetic technology is sufficiently explicitable that various software packages can do many computations both for algebraic geometry (and commutative algebrac) and arithmetic algebraic geometry (and algebraic number theory), in particular this is the case with elliptic curves and modular forms. This is a recent developpment (and it is due to the amazing work of many people), and it can be exploited by students to get more intuition about all these theories.

(One exception seems to be étale cohomology where little is apparently computable, as witnessed by the enormous amount of work required in the work of Couveignes, Edixhoven, and otthers, to compute the Ramanujan tau function at prime arguments using Deligne’s results that show where it can be found in étale land).

9 September, 2008 at 1:35 pm

Przemyslaw ChojeckiI also would like to thank very much Matthew Emerton. The advice given above are of great use to me and are spoken just in time for me (just when I started to touch delicately the vast surface of arithmetic algebraic geometry).

My general thought is, it would be absolutely amazing and useful for many students if there will be a single place (a webpage), where experts would give their insights into their field of expertise, tell how to start and what is worth of reading at the beginning to jump into a field. The short note of lenght as that of M. Emerton is more than enough.

Best Regards

16 March, 2009 at 12:29 pm

sheavesDear Prof. Emerton,

Thank you for your advice! I have read them and bookmarked this page!

I have one question, though. I am planning on doing a PhD in abstract algebraic geometry (as of now, I only have a vague idea of what abstract AG is but I tend to go with the rule: the more abstract, the better) rather than arithmetic algebraic geometry. I am woundering if you still advice not to jump directly to EGA or SGA after having done Hartshorne.

I guess Serre’s paper on coherent sheaves is a good thing to read too.

Thank you.

16 March, 2009 at 7:19 pm

Matthew EmertonDear Sheaves,

Serre’s paper on coherent sheaves is certainly great. There is also a very nice survey on sheaf cohomology on algebraic geometry written by Zariski. (It appeared in the Bulletin of the AMS in the late 50’s, I think, as part of the proceedings of a workshop on algebraic geometry — you should be able to find it on MathSciNet. It is also in his collected works; volume 4, I think.)

Zariski himself never used sheaves or cohomology; his work belongs to an earlier era. But in his survey he carefully explains the basics of sheaf cohomology, but more importantly, he illustrates its power as a tool by discussing several examples of geometric theorems of himself and others which are easily established via cohomological arguments. (One good example that he gives explains why in Hartshorne a lemma about the vanishing of H^1 in certain contexts is called the lemma of Enriques-Severi-Zariski, when none of these three geometers ever worked with a cohomology group.)

I think that especially if you want to work in a general and abstract setting, this survey might be useful, but it will help give you a sense of what certain ideas mean geometrically, which can be hard to intuit from the more algebraic formalism of algebraic geometry as presented in Hartshorne.

Another paper you might want to read is Serre’s GAGA paper. In this paper one sees how a rather abstract argument based on sheaf cohomology allows one to deduce concrete results such as Chow’s theorem, stating that any complex analytic subvariety of P^n(C) is actually an algebraic subvariety.

As to how much time to spend on EGA and SGA, this is something that you ultimately have to decide for yourself, hopefully with the guidance of your thesis adviser. But one thing to remember is that many very clever people have pored over the details of EGA and SGA for many years now, and it so it is going to be hard for anyone to find interesting new results that can be obtained just by applying the ideas from these sources alone (as important as those ideas are). Even if you want to make progress in a very general, abstract setting, you will need ideas to come from somewhere, motivated perhaps by some new phenomenon you observe in geometry, or number theory, or arithmetic, or … .

This is part of the reason why I advise against spending too much time just holed up with EGA and SGA. By themselves, they are not likely (for most people) to provide the inspiration for new results. On the other hand, when you are trying to prove your theorems, you might well find techincal tools in them which are very helpful, so it is useful to have some sense of what is in them and what sort of tools they provide. But you will likely have to find your inspiration elsewhere, and so for this reason alone you will probably want to make time to look at other things too.

Regards,

Matthew

16 March, 2009 at 7:21 pm

Matthew EmertonCorrection of a typo: “but it will help give you a sense” should read “because it will help give you a sense”.

17 May, 2009 at 10:42 am

JonDear Professor Tao,

Thank you for all of the information on this blog. It is a very inspiring and useful source for young mathematicians everywhere.

I imagine it is accurate to say that learning and relearning your field is easier to do if it can be done while making progress, i.e. proving publishable results along the way. How does one do this most effectively?

I mean, it certainly would be great for one’s edification to rewrite a classic text in one’s field, but this may be crazy if one doesn’t obtain new results to publish in the process…especially as a text I write will probably not be well-received.

I guess that my questions are, how should one evaluate for payoff? How much of one’s time should be devoted to writing expository things? Should it be that we write an expository work and “spin-off” papers while writing it? I guess I am asking if this is the way you work. I personally feel guilt and pressure writing expository things, as when I am doing this I usually am not writing papers. I wonder if refining my exposition of the basic areas of my field might help produce better papers, though.

Anyhow, thanks for everything!

Jon

19 May, 2009 at 6:12 pm

Terence TaoDear Jon,

There is of course a tradeoff between the short-term goal of writing one’s next paper, and the long-term goal of learning more about one’s field (which, presumably, one would be writing dozens of future papers in). I personally like to alternate between these goals: I would work on a research paper if I feel particularly inspired to do so, but then fall back to more general tasks, such as learning some relevant piece of mathematics properly, when I feel I need a change.

It’s not particularly efficient to go and try to rewrite a textbook, work out an enormous number of exercises, or digest a huge topic unrelated to your existing research, though; it is preferable to have a specific, feasible, and research-oriented goal in mind when trying to learn something. For instance, in one of your papers you may be using method X to achieve a certain key step in the argument, but in other papers in the literature method Y (which you don’t know very well) is used instead to obtain similar, but subtly different results. It then makes sense to learn about Y with the intent of comparing it with X, for instance trying to translate a simple proof using Y into a proof using X, or vice versa, figuring out (at least heuristically) what types of results are achievable using Y but not X or vice versa, or seeing to what extent X and Y can be combined. Ideally, once you work these things out, then you will be automatically be able to judge whether X, Y, or something else is the best tool for the job whenever a similar task comes up again in your research.

17 May, 2009 at 10:44 am

JonTo sharpen my post a bit:

How does one get to the point where one feels that expository writing is integrated with one’s research and not just procrastination or additional (and fruitless) effort?

Jon

22 May, 2009 at 7:37 am

JonYour response was very helpful. On a related note about learning and relarning one’s field, I think readers of the blog would be interested in your advice or view on the “Moore Method” debate.

Some mathematics professors think that one should only work in areas in which one has, without hints, proved ALL of the basic results of the area needed to begin research. (Much like how R.L. Moore ran his courses training graduate students in general topology.) Along these lines, Paul Halmos supposedly said that one should “never read the proof of a theorem”, but should instead simply use the result if one cannot prove it himself.

Opposed to this, it seems unrealistic to never read any proofs in order to gain new techniques. One of Halmos’s former students told me that he tries to prove a theorem (in a given paper) for himself, and then when stuck takes hints from the paper he is reading until he proves everything “with hints”. This person says, though, that he still doesn’t read enough to keep up with current developments.

I guess I’d like to know, what’s your take on the Moore Method?

Thanks,

Jon

29 May, 2009 at 3:20 am

JonIn retrospect, asking the above question on the Moore Method is moot, as you clearly encourage learning new techniques from the literature. I asked the Moore question because I think that a word from you on the subject would have enough sway to encourage or discourage the method in the training of students. This is, now that I think about it, politically, a very unfair question to ask! (My curiosity simply got the better of me.) My sincere apology on this count!

Thanks again for your advice above.

Best,

Jon

30 May, 2009 at 2:15 am

JonFor any who are interested, I think that Professor Tao’s post above can be very useful for advising students (graduate students?) on how to read mathematics. Perhaps one should read arguments that you feel will (in some sense) subtly differ from an argument you already have constructed. I wonder if this may apply to undergraduate students as well, since the advice seems useful even if the argument you have constructed is wrong. (This reminds me of Gowers’s statement that mathematicians are always trying to imagine arguments “no matter how vague”). This may be a nice compromise on the Moore method, and may provide students with a productive maxim on mathematical reading.

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Work hard[…] goes, and then ultimately writing up a complete and detailed treatment of the topic. (See also “learn and relearn your field“.) It would be very pleasant if one could just dream up the grand ideas and let some “lesser […]

14 December, 2011 at 2:52 pm

“La matemática es más que rigor y demostraciones” | blocdemat[…] El punto del rigor no es destruir la intuición; al contrario, debería ser usada para destruir la mala intuición al clarificar y elevar la buena intuición. Sólo es con una combinación tanto del formalismo riguroso y la buena intuición que uno puede atacar problemas matemáticos complejos; uno necesita del primero para tratar correctamente con los detalles finos, y de la segunda para tratar correctamente con el panorama general. Sin uno o el otro, estarás mucho tiempo dando tumbos en la oscuridad (lo cual puede ser instructivo, pero de una gran ineficiencia). Entonces una vez que estés plenamente cómodo con el pensamiento matemático riguroso, deberías revisitar tus intuiciones sobre el tema y usar tus nuevas habilidades de pensamiento para poner a prueba y refinar estas intuiciones en vez de descartarlas. Una manera de hacer esto es “ask yourself dumb questions“; otra es “relearn your field“. […]

6 September, 2012 at 1:46 am

O projektu MatemaTech | Matematech[…] NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the […]

29 November, 2012 at 4:16 pm

EllenEnsherYou can use this philosophy in any field you are working in as well, not just mathematics.

Inspirational post! :)

3 December, 2012 at 7:06 am

Anonymousdear prof tao

do you read classical papers too? and do you recommend reading them? if yes in which stage? undergraduate? graduate? …

9 December, 2012 at 2:25 am

[Skills] Làm việc chăm chỉ – GS Terrence Tao | Nguyen Hoai Tuong[…] nghĩ là mình hiểu về một bài toán nào đó thì bạn phải chắc rằng mình đã đọc tất cả các tài liệu liên quan, viết ra ít nhất một phác thảo về nó, và viết ra được cách giải hoàn […]

27 March, 2013 at 11:42 pm

Learn and relearn your field -Terence Tao | Readings for the Distinguishing Palatte[…] https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/ […]

15 April, 2013 at 12:12 am

我的动机们 | Shrinklemma[…] 想把分析qual全做一遍的动机：对自己的分析功底不够自信（这就是一个靠不住的动机，因为它不是一个纯粹的动机，它甚至是一个荒谬的动机，因为你永远不会对你的分析功底真正“自信”。陶写过博客Learn and relearn your field，其中就是在说你要随时重新学习你已经学过的东西，换句话说，从已知的材料里，尝试挖掘所有可能挖掘出来的东西。所以，做分析qual里的题目，按照动机来论，落入relearn your field这一个框架里，对于这个动机，做qual的题目，不如写notes，尝试自己把理解的东西写下来，然后做更多的补充，关于这一点，更要参考陶的一篇博客Write down what you’ve done。至于做qual的题目，可以把他归于“brain is a muscle, need practising”这样一个动机里。） […]

21 May, 2013 at 3:43 am

Bisogna essere un genio per fare matematica? - Maddmaths[…] NO enfatico. Per dare dei contributi buoni ed utili alla matematica, uno deve lavorare duramente, conoscere bene un settore, conoscere altri settori e altri strumenti, fare domande, parlare con altri matematici e pensare al […]

19 July, 2013 at 8:57 am

Does one have to be a genius to do maths? | Singapore Maths Tuition[…] NO. In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fields and tools, ask questions, talk to other mathematicians, and think about the […]

24 October, 2013 at 9:47 am

Terry Tao: On Hard Work | Fahad's Academy[…] goes, and then ultimately writing up a complete and detailed treatment of the topic. (See also “learn and relearn your field“.) It would be very pleasant if one could just dream up the grand ideas and let some “lesser […]

25 June, 2014 at 7:39 pm

افضل شركة رش مبيدات بالرياضWhen some one searches for his vital thing, so he/she needs to be available that in detail, thus that thing is maintained over here.

17 October, 2014 at 11:06 am

Hay que dar un tiempo para que las ideas se asienten | Didáctica de la Filosofía[…] me comenta que en el blog de Terry Tao hay una entrada (más enfocada a las matemáticas) donde se recomienda aprender las cosas una y otra vez. La moraleja es que tanto los profesores como los estudiantes han de tener paciencia. Lo […]

6 February, 2015 at 5:27 pm

Career Advice by Prof Terence Tao, Mozart of Mathematics | MScMathematics[…] one. Later in your research career, you will find that problems are mainly solved by knowledge (of your own field and of other fields), experience, patience andhard work; but for the type of problems one sees […]

11 July, 2015 at 5:57 am

译：解决数学问题 by 陶哲轩 | 万里风云[…] 解决问题，无论是对作业题还是对人类未曾解决的问题，当然都是数学学科的重要方面，尽管这并不是唯一的方面。在今后你的研究生涯里，你会发现，人们往往通过知识（包括你自己领域和别的领域的知识）、经验、耐心、和勤奋工作来解决问题；但是对于中小学、大学、或者数学竞赛里的题目，人们需要一套稍微与众不同的问题解决技巧。我写过一整本关于在这个层面上解决数学题的书；而且，那本书的第一章就在讨论通用的问题解决策略。当然，市面上也有很多其他关于解决问题的书，比如波利亚的《怎样解题》——我自己在准备数学奥林匹克竞赛时，就在学习这本书。 […]

24 December, 2015 at 1:02 pm

Review: A PhD Is Not Enough | The Intellectual Journal of Daniel Ang[…] Tao’s career advice, especially the importance of learning and relearning your field. (This is meant for mathematicians but the general philosophy and mentality can be transferred […]

9 May, 2016 at 2:54 pm

alexI’ve always tried to understand deeply every idea in methematic lectures but sometimes i find myself stuck with problems that are not quite related to the subject that im reading about ;and solving or proofing them takes relatively a lot of time ,so my question is: can i only focus my regourous thinking on the main ideas and just try to understand intuitively the secondary problems ?

22 June, 2016 at 1:10 am

Smart Imposter Syndrome – G-Notes[…] In order to make good and useful contributions to mathematics, one does need to work hard, learn one’s field well, learn other fieldsand tools, ask questions, talk to other mathematicians, and think about the […]

18 July, 2016 at 8:29 am

Solving mathematical problems | nguyen Huynh Huy's Blog[…] one. Later in your research career, you will find that problems are mainly solved by knowledge (ofyour own field and of other fields), experience, patience and hard work; but for the type of problems one sees in […]

13 August, 2016 at 4:18 am

Random Stuff « Econstudentlog[…] rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field.” (Terry Tao, There’s more to mathematics than rigour and […]

23 January, 2018 at 8:24 pm

Có Nhiều Hơn là Sự Chặt Chẽ và Chứng Minh trong Toán Học | 5[…] Ý nghĩa của sự nghiêm ngặt không nằm trong việc tiêu diệt toàn bộ cảm tính; thay vào đó, nó nên được sử dụng để tiêu diệt các cảm tính xấu và đồng thời làm rõ và nâng cao các cảm tính tốt. Chỉ với sự phối hợp giữa sự định nghĩa chặt chẽ và cảm tính tốt thì một nhà nghiên cứu mới có thể tấn công các bài toán phức tạp; người này cần cái thứ nhất để có thể giải quyết các chi tiết tinh tế, và cần cái sau để có thể xử lý được bức tranh toàn cục của vấn đề. Chỉ thiếu một trong hai điều trên, bạn sẽ bỏ rất nhiều thời gian mò mẫm trong bóng tối (điều này có thể hữu ích, nhưng cực kỳ không hiệu quả). Thế nên khi bạn đã hoàn toàn thoải mái với suy luận toán học chặt chẽ, bạn nên thăm lại các cảm tính của mình thay vì vứt bỏ chúng. Một cách để làm điều này là tự hỏi bản thân các câu hỏi ngớ ngẩn; một cách khác là học lại phân ngành của bạn. […]

13 July, 2018 at 7:20 pm

Five methods to learn effectively – puzzlr[…] Further reading Original source Feynman’s organisation of knowledge The thoughts of Terence Tao. […]

24 March, 2019 at 5:14 pm

Review: A PhD Is Not Enough – Daniel Ang[…] Tao’s career advice, especially the importance of learning and relearning your field. (This is meant for mathematicians but the general philosophy and mentality can be transferred […]

3 April, 2019 at 6:20 am

emerton – 站点标题[…] https://terrytao.wordpress.com/career-advice/learn-and-relearn-your-field/#comment-32483 […]

10 July, 2020 at 4:48 am

Newsletter NSSN nº8 – Not So Short Notes[…] Learn and relearn your field, by Terence Tao […]

2 August, 2020 at 8:38 pm

Imran HussainThanks professor for making these wonderful advices available on your blog .Hope you will continue to write such advices.

11 August, 2020 at 12:11 am

Hani fathimaHello Professor,

I am newly PhD scholar who is interested to learn and relearn Mathematics.It would be a great help if you provide some tips on how should i work on my Thesis.What are the books should i read before i start my PhD?

Thank You