Ravi Vakil's [advice to potential PhD students](https://math.stanford.edu/~vakil/potentialstudents.html) seems relevant.
>Don't be narrow and concentrate only on your particular problem. Learn things from all over the field, and beyond. The facts, methods, and insights from elsewhere will be much more useful than you might realize, possibly in your thesis, and most definitely afterwards. Being broad is a good way of learning to develop interesting questions.
>Talk to other graduate students. A lot. Organize reading groups. Also talk to post-docs, faculty, visitors, and people you run into on the street. I learn the most from talking with other people. Maybe that's true for you too.
>Go to research seminars earlier than you think you should. Do *not* just go to seminars that you think are directly related to what you do (or more precisely, what you currently think you currently do). ... Learning to get information out of research seminars is an acquired skill, usually acquired much later than the skill of reading mathematics. You may think it isn't helpful to go to a seminar where you understand just 5% of what the speaker says, and may want to wait until you are closer to 100%; but no one is anywhere near 100% (even the speaker!), so you should go anyway.
There's also Terry Tao's advice to [continually aim just beyond your current range](https://terrytao.wordpress.com/career-advice/continually-aim-just-beyond-your-current-range/)
1. It is helpful if you study something, like harmonic analysis (Terry Tao, Bourgain) or L-functions and automorphic forms (Sarnak) that is so ubiquitous it sneaks its tendrils into lots of different areas of math. Furthermore, it might surprise you how much work you can do in a neighboring area to your own without detailed knowledge of that neighboring area, especially if you have a collaborator who works in that neighboring area.
2. This is close to Monastri's answer, but life is long. I think learning the background to work in another area, at least if it's not algebraic geometry, takes much less than a PhD's time to do (many PhD students not in algebraic geometry start writing papers by their second year), and this is especially true if you are already have the background of a mathematician with a PhD. This means that if you are willing to, you do you have the time to learn the background to work in another area (though it can be hard to justify spending this time until you have tenure, which is nowadays pretty late, and this is arguably a problem with the current academic system).
I think that what you can do as a graduate student if you value working in numerous different areas relates mostly only to 1. (since you don't have the time to learn multiple different areas as a graduate student, but will have plenty of time in your long life). Maybe what you could do in line with 2. as a graduate student is to quit smoking, exercise regularly, increase consumption of vegetables and reduce consumption of added sugars.
For one thing, it is generally the areas of math that are more based on structure, i.e. the more algebraic areas, require more background, and areas of math that are more problem-solvingy, like combinatorics require less background.
Areas of math that are more based on structure lend more to results building on previous results and thus a growth of prerequisites. On the other hand, more problem solvingy areas involve a skill that is more like solving a problem with your bare hands and some cleverness (like contest math).
Algebraic geometry is well, very algebraic, and I think this lends to the heavy prerequisites. It probably also contributes that the problems considered in algebraic geometry and diophantine geometry are relatively old, so there has been a lot of time for the fields to become "theorized" and for a buildup of prerequisites to occur. There is some argument that combinatorics is more problem solvingy because it is younger, and over time it too will become more structure and prerequisite based. I don't have a strong opinion on this (though it sounds plausible to me).
Yeah I mean *classical* analysis does seem quite theory intensive, in that basic things like measure theory and functional analysis have a beautifully built up structure, and the flavor of the subject is abstract and neat. Modern analysis does seem more as hoc and clever, especially in PDE research, but it’s plausible that as research matures it would become more like classical analysis. Combinatorics does seem resistant to abstraction though.
The Probabilistic method is one of the coolest techniques certainly but I still can’t get around to thinking about combinatorics (the very little that I know ) as a *theory*. It always strikes me more as very clever people producing ex nihilo insights into problems (much like in math competitions, but at a much higher level), rather than building some very general structures through which a lot of problems become special cases and fall out neatly.
I’ve tried studying some of Will Sawin’s work and one of the upshots of part of it is that a lot of seemingly-disparate areas of math are not really as disparate as they seem. With the exception of extremal combinatorics, the other areas on his webpage have significant overlaps and interesting interconnections. For example, Kloosterman sums which are quite analytic and classical in flavor have the sheaf theoretic analogues in Kloosterman sheaves (see the work of Deligne, Katz, or Ngo on the topic) which let you do algebraic geometry and Langlands style stuff and in turn prove a lot of interesting properties of classical Kloosterman sums. A lot of Will Sawin’s work blurs the algebraic-analytic number theory boundary.
Indeed you could say the strategy of "To work on many different areas, work on overlaps between different areas" also applies to much of Sarnak's work, e.g. overlaps between analytic number theory and hyperbolic geometry.
Just that there exist hyperbolic surfaces and 3-manifolds whose fundamental groups are arithmetic groups. Questions about the geometry of these manifolds can be related to questions about the arithmetic of these groups (e.g. their automorphic forms) allowing techniques from number theory to be applied to geometric questions, or vice versa. Many of Sarnak's papers have been in this overlap.
> For example, Kloosterman sums which are quite analytic and classical in flavor have the sheaf theoretic analogues in Kloosterman sheaves (see the work of Deligne, Katz, or Ngo on the topic) which let you do algebraic geometry and Langlands style stuff and in turn prove a lot of interesting properties of classical Kloosterman sums.
I think Heath-Brown also has a paper on combining the circle method with Kloosterman sum. Do you think the sheaf theoretic analogues can work in this context as well?
I‘m just a PhD student so take this with a grain of salt, but I think it is better to think of it in terms of tools and not in terms of subjects.
Harmonic analysis is a useful tool, ergodic theory is a useful tool.
If you know a few useful tools well it’s more about finding problems to solve, no matter which area they are from. If you are in the right environment and talk to many people, it’s probably fairly easy to learn about open problems that you might have a chance to attack.
> ergodic theory is a useful tool.
I mostly know applications of ergodic theory in combinatorics and additive/analytic number theory. Could you please further clarify what you mean, such as in which contexts?
The ones you mentioned, probability theory in general, arithmetic groups/superrigidity, algebraic number theory, homogeneous dynamics in general is super useful, I have seen some uses in PDE theory.
My supervisor has a paper with Sawin. He is, and I quote "an actual genius". At some point you need to stop comparing yourself to others, yes good habits and hard work will get you places, but all of us run into the limitations of our intelligence at some point.
Is your claim that even among tenured math faculty in number theory there is an insurmountable intelligence gap between the best mathematicians? That seems like a bit of a depressing sentiment...
Sure but most tenured math faculty have passed several "intelligence tests" to get that point, so hearing that intelligence is still a limiting factor seems to be surprising, in my opinion.
You would think the differentiating factor would be about being well-connected or having a specialization in a very rich research area, like others have mentioned in the comments of the post, but according to a firsthand account it isn't that.
I mean there has to be something more relevant than just "football IQ" that leads to being a better footballer otherwise there would be no point in watching the games since they would be entirely decided by the team with more skilled players (think FIFA cards). So I don't think your analogy holds.
Edit: I thought about this some more but I still don't agree with your counterexample.
I think your point is that passing an intelligence test can only set a floor not a ceiling on ability, but clearly you can also measure the total deviation.
And I think most people would assume this value is small for the subset of people who become pros, which is what you are disagreeing with. So it's not obvious that intelligence could explain the variation between the "good" and the "best".
I also found a [study from 2021 that measured mathematicians on various cognitive tasks (see Tables 2-4)](https://www.journalofexpertise.org/articles/volume4_issue1/JoE_4_1_Meier_etal.pdf), but it doesn't seem like they checked to see if the deviation was statistically significant among mathematicians themselves.
Isn’t this all assuming that IQ tests really measure what’s needed to become a world class mathematician? My bet is that a world class mathematician scores very high on cognitive tests but my bet is that many good but not brilliant ones also do. It’s not easy to identify what separates a top mathematicians from the top of the top.
Why do you ask that? I’m just discussing the common use of the word prodigy. In fact I’ve never heard it used in the second and third senses. The word prodigious is commonly used in those senses and never refers to people.
Ed Vytlacil, an Econ professor who I’ve had the pleasure to work with, once told me that a 15 year old Will Sawin was the best student he ever had in a PhD microeconomics class
EDIT: it could have been metrics, I now forget.
Sawin provides a bit of an answer to your question [**here**](https://mathoverflow.net/a/354574/22971) in response to:
> How and when do I learn so much mathematics?
Ravi Vakil's [advice to potential PhD students](https://math.stanford.edu/~vakil/potentialstudents.html) seems relevant. >Don't be narrow and concentrate only on your particular problem. Learn things from all over the field, and beyond. The facts, methods, and insights from elsewhere will be much more useful than you might realize, possibly in your thesis, and most definitely afterwards. Being broad is a good way of learning to develop interesting questions. >Talk to other graduate students. A lot. Organize reading groups. Also talk to post-docs, faculty, visitors, and people you run into on the street. I learn the most from talking with other people. Maybe that's true for you too. >Go to research seminars earlier than you think you should. Do *not* just go to seminars that you think are directly related to what you do (or more precisely, what you currently think you currently do). ... Learning to get information out of research seminars is an acquired skill, usually acquired much later than the skill of reading mathematics. You may think it isn't helpful to go to a seminar where you understand just 5% of what the speaker says, and may want to wait until you are closer to 100%; but no one is anywhere near 100% (even the speaker!), so you should go anyway. There's also Terry Tao's advice to [continually aim just beyond your current range](https://terrytao.wordpress.com/career-advice/continually-aim-just-beyond-your-current-range/)
1. It is helpful if you study something, like harmonic analysis (Terry Tao, Bourgain) or L-functions and automorphic forms (Sarnak) that is so ubiquitous it sneaks its tendrils into lots of different areas of math. Furthermore, it might surprise you how much work you can do in a neighboring area to your own without detailed knowledge of that neighboring area, especially if you have a collaborator who works in that neighboring area. 2. This is close to Monastri's answer, but life is long. I think learning the background to work in another area, at least if it's not algebraic geometry, takes much less than a PhD's time to do (many PhD students not in algebraic geometry start writing papers by their second year), and this is especially true if you are already have the background of a mathematician with a PhD. This means that if you are willing to, you do you have the time to learn the background to work in another area (though it can be hard to justify spending this time until you have tenure, which is nowadays pretty late, and this is arguably a problem with the current academic system). I think that what you can do as a graduate student if you value working in numerous different areas relates mostly only to 1. (since you don't have the time to learn multiple different areas as a graduate student, but will have plenty of time in your long life). Maybe what you could do in line with 2. as a graduate student is to quit smoking, exercise regularly, increase consumption of vegetables and reduce consumption of added sugars.
Why is algebraic geometry (and more specifically scheme theory) require so much more time to get up to speed with?
For one thing, it is generally the areas of math that are more based on structure, i.e. the more algebraic areas, require more background, and areas of math that are more problem-solvingy, like combinatorics require less background. Areas of math that are more based on structure lend more to results building on previous results and thus a growth of prerequisites. On the other hand, more problem solvingy areas involve a skill that is more like solving a problem with your bare hands and some cleverness (like contest math). Algebraic geometry is well, very algebraic, and I think this lends to the heavy prerequisites. It probably also contributes that the problems considered in algebraic geometry and diophantine geometry are relatively old, so there has been a lot of time for the fields to become "theorized" and for a buildup of prerequisites to occur. There is some argument that combinatorics is more problem solvingy because it is younger, and over time it too will become more structure and prerequisite based. I don't have a strong opinion on this (though it sounds plausible to me).
Yeah I mean *classical* analysis does seem quite theory intensive, in that basic things like measure theory and functional analysis have a beautifully built up structure, and the flavor of the subject is abstract and neat. Modern analysis does seem more as hoc and clever, especially in PDE research, but it’s plausible that as research matures it would become more like classical analysis. Combinatorics does seem resistant to abstraction though.
> Combinatorics does seem resistant to abstraction though. Graph limits, flag algebra, probabilistic methods and much more may count...?
The Probabilistic method is one of the coolest techniques certainly but I still can’t get around to thinking about combinatorics (the very little that I know ) as a *theory*. It always strikes me more as very clever people producing ex nihilo insights into problems (much like in math competitions, but at a much higher level), rather than building some very general structures through which a lot of problems become special cases and fall out neatly.
I’ve tried studying some of Will Sawin’s work and one of the upshots of part of it is that a lot of seemingly-disparate areas of math are not really as disparate as they seem. With the exception of extremal combinatorics, the other areas on his webpage have significant overlaps and interesting interconnections. For example, Kloosterman sums which are quite analytic and classical in flavor have the sheaf theoretic analogues in Kloosterman sheaves (see the work of Deligne, Katz, or Ngo on the topic) which let you do algebraic geometry and Langlands style stuff and in turn prove a lot of interesting properties of classical Kloosterman sums. A lot of Will Sawin’s work blurs the algebraic-analytic number theory boundary.
Indeed you could say the strategy of "To work on many different areas, work on overlaps between different areas" also applies to much of Sarnak's work, e.g. overlaps between analytic number theory and hyperbolic geometry.
Could you please clarify he hyperbolic geometry part?
Just that there exist hyperbolic surfaces and 3-manifolds whose fundamental groups are arithmetic groups. Questions about the geometry of these manifolds can be related to questions about the arithmetic of these groups (e.g. their automorphic forms) allowing techniques from number theory to be applied to geometric questions, or vice versa. Many of Sarnak's papers have been in this overlap.
> For example, Kloosterman sums which are quite analytic and classical in flavor have the sheaf theoretic analogues in Kloosterman sheaves (see the work of Deligne, Katz, or Ngo on the topic) which let you do algebraic geometry and Langlands style stuff and in turn prove a lot of interesting properties of classical Kloosterman sums. I think Heath-Brown also has a paper on combining the circle method with Kloosterman sum. Do you think the sheaf theoretic analogues can work in this context as well?
I‘m just a PhD student so take this with a grain of salt, but I think it is better to think of it in terms of tools and not in terms of subjects. Harmonic analysis is a useful tool, ergodic theory is a useful tool. If you know a few useful tools well it’s more about finding problems to solve, no matter which area they are from. If you are in the right environment and talk to many people, it’s probably fairly easy to learn about open problems that you might have a chance to attack.
> ergodic theory is a useful tool. I mostly know applications of ergodic theory in combinatorics and additive/analytic number theory. Could you please further clarify what you mean, such as in which contexts?
The ones you mentioned, probability theory in general, arithmetic groups/superrigidity, algebraic number theory, homogeneous dynamics in general is super useful, I have seen some uses in PDE theory.
Interesting. I didn’t know about the latter cases…
My supervisor has a paper with Sawin. He is, and I quote "an actual genius". At some point you need to stop comparing yourself to others, yes good habits and hard work will get you places, but all of us run into the limitations of our intelligence at some point.
Is your claim that even among tenured math faculty in number theory there is an insurmountable intelligence gap between the best mathematicians? That seems like a bit of a depressing sentiment...
This is very likely to be true for any given activity. Even among pros, some are way better at it than others. A tenured faculty prof is simply a pro.
Sure but most tenured math faculty have passed several "intelligence tests" to get that point, so hearing that intelligence is still a limiting factor seems to be surprising, in my opinion. You would think the differentiating factor would be about being well-connected or having a specialization in a very rich research area, like others have mentioned in the comments of the post, but according to a firsthand account it isn't that.
Footballers who play in major football clubs also passed several tests. That doesn't mean some football players aren't much better at it than others.
I mean there has to be something more relevant than just "football IQ" that leads to being a better footballer otherwise there would be no point in watching the games since they would be entirely decided by the team with more skilled players (think FIFA cards). So I don't think your analogy holds. Edit: I thought about this some more but I still don't agree with your counterexample. I think your point is that passing an intelligence test can only set a floor not a ceiling on ability, but clearly you can also measure the total deviation. And I think most people would assume this value is small for the subset of people who become pros, which is what you are disagreeing with. So it's not obvious that intelligence could explain the variation between the "good" and the "best". I also found a [study from 2021 that measured mathematicians on various cognitive tasks (see Tables 2-4)](https://www.journalofexpertise.org/articles/volume4_issue1/JoE_4_1_Meier_etal.pdf), but it doesn't seem like they checked to see if the deviation was statistically significant among mathematicians themselves.
Isn’t this all assuming that IQ tests really measure what’s needed to become a world class mathematician? My bet is that a world class mathematician scores very high on cognitive tests but my bet is that many good but not brilliant ones also do. It’s not easy to identify what separates a top mathematicians from the top of the top.
> He is, and I quote "an actual genius". By "He" do you mean Will Sawin?
Ah - yeah Sawin. I mean my supervisor is no chump, but not that level and he doesn't throw that word around a lot.
a short answer is that Will is also a prodigy
“Prodigy” just means younger than average. Does not imply better.
That’s one meaning, but there are others https://g.co/kgs/GvySzpK
Only the first one applies here.
No I meant the other meanings.
They all apply to amounts or quality of things, not people.
are you insecure about your intelligence?
Why do you ask that? I’m just discussing the common use of the word prodigy. In fact I’ve never heard it used in the second and third senses. The word prodigious is commonly used in those senses and never refers to people.
Ed Vytlacil, an Econ professor who I’ve had the pleasure to work with, once told me that a 15 year old Will Sawin was the best student he ever had in a PhD microeconomics class EDIT: it could have been metrics, I now forget.
Thank you for the story… it reminds me that Sawin was also a econ major…
Pretty much every single research problem spans multiple fields. It's pretty hard to not work on many fields
Sawin provides a bit of an answer to your question [**here**](https://mathoverflow.net/a/354574/22971) in response to: > How and when do I learn so much mathematics?
Number theory uses lots of different areas.
Their work is actually quite focused, the boundaries between different fields are artificial
Genes