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$\DeclareMathOperator\GL{GL}$Many times I have heard people say sentences like X is an important question/ X is a natural question. I find this very surprising because to me it's all a matter of taste. I am having people ask me why study certain things and my mental response is it's fun or just out of curiosity; but often what I found is when I present my question with enough jargon then they agree my questions are worth studying.

An example is Schur positivity: for me it's an extremely rare phenomenon and every time a family of symmetric functions is Schur positive I feel it's worthy of study in its own right. But when I need to explain to people I need to talk in terms of Frobenius map, representations of $\GL_n$ and so on. But I never understood myself why representations of symmetric group or $\GL_n$ is more important than symmetric functions.

So I really want to know how to decide whether a question is worth studying? How do I decide what question is important to ask in mathematics?

I am sorry if this is not the right place to ask this. I will remove it if it violates MO policy somehow.

LSpice
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ArB
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    This is not an easy question. You may wish to have a look at Tao, Terence, What is good mathematics? Bull. Am. Math. Soc., New Ser. 44, No. 4, 623-634 (2007). – Matthieu Romagny Feb 24 '23 at 08:47
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    I disagree quite strongly with the reason for the close vote. Of course a significant degree of opinion necessarily occurs here, but there are also quite a lot of things to say from a more objective perspective (at least at a meta-level, i.e. "what makes people consider a question to be important?"). – Jochen Glueck Feb 24 '23 at 08:57
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    I mean, seriously, the entire academic ecosystem is built on reputation, which relies in turn to a significant degree on perceived relevance of research questions and results. Whether your research is seen as important decides whether you publish in the Annals or in Rejecta Mathematica, whether you become a professor or Dr tried-hard-but-couldn't-find-a-postdoc. Careers and large amounts of money depend on what is considered to be important - but when somebody asks "Ok, so how do you determine what is important?", then it's suddenly entirely opinion-based? – Jochen Glueck Feb 24 '23 at 08:57
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    There is a view that natural problems are bad problems https://gilkalai.wordpress.com/2019/04/25/are-natural-mathematical-problems-bad-problems/ – Gil Kalai Feb 24 '23 at 10:02
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    I wrote some of my thoughts on the mechanisms of what makes interesting or fashionable research in this answer. But this question is slightly different, and I would be interested to read people's ideas. – R. van Dobben de Bruyn Feb 24 '23 at 13:08
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    Here is a perhaps silly definition: a question X is important at time t if the resolution of X allows the solution to many acknowledged open questions in the literature at time t. (This is surely insufficient...but is it right to zeroth order, even?) – Jon Bannon Feb 24 '23 at 14:43
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    This is a question I have wanted to ask but assumed off topic. I hope it stays open. – JP McCarthy Feb 24 '23 at 15:45
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    Important mathematical questions are questions raised by Important mathematicians. Important mathematicians are mathematicians who answer important questions :) – Benjamin Steinberg Feb 24 '23 at 16:52
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    @JochenGlueck, your narrative is a little idealistic, eh? It's the interplay between what you know and who you know that establishes viable careers. I'm sure many have examples of this. One of my favorites is Gian-Carlo Rota starting the journal Advances in Mathematics to publish his own papers, which allowed him to build a medium via which to establish a mutual-support group of individuals who could then form a consensus on what questions were 'important' to them, or central to 'progress' in their shared interests, in the fields of combinatorics and probability. Ex. 2) Abel and A. Crelle. – Tom Copeland Feb 24 '23 at 17:22
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    Ex. 3) The Scottish Cafe and the Scottish Book (https://en.wikipedia.org/wiki/Scottish_Caf%C3%A9). – Tom Copeland Feb 24 '23 at 17:26
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    @TomCopeland: Idealistic? Well, that's certainly none of the word's that I had in mind when I wrote my comments above. I'm also not quite sure where you see a "narrative". Perceived importance of one's research is of course not the only major influence on one's careers - but it certainly is one point that has a very strong influence. – Jochen Glueck Feb 24 '23 at 17:34
  • I think this is a very important question. Clearly some questions are more important in math then others. Sometimes it is a mode of operation in which motive is not important. – marshal craft Feb 24 '23 at 17:39
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    I think In my opinion an important question is one that reveals a lot. So if you compared Fermat last theorem to say Galoui theory, I’d say Fermats theorem is not as important because it does not reveal as much. It may be more complex but is not more significant. I’d say too there is a tendency today for mathematics to be more complex while less significant. There has been a lot of discoveries and because of that there is a lot to explore and problems which arise while doing that. – marshal craft Feb 24 '23 at 17:51
  • And as you ask all of this can be made concrete, you can explicitly state these things. – marshal craft Feb 24 '23 at 17:53
  • There are some questions that are significant or maybe not too, like p!=np if it’s true it doesn’t help much but it remains unproven. Obv if it were false it would be important cause it would improve computation times which would be useful. – marshal craft Feb 24 '23 at 17:56
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    @marshalcraft, of course, "as it happened", the resolution of Fermat's Last Theorem by Wiles (with help from Taylor) involved partial proof of the Shimura-Taniyama conjecture, which (to me) was something I thought I'd never see! :) – paul garrett Feb 24 '23 at 19:22
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    @JochenGlueck, re, I'm not quite sure what to conclude from your comment. As you point out, how it is decided whether a problem is important is itself important, and needs to be known. But there are a lot of important things that are largely, if not wholly, opinion based—and I'd argue that reputation and importance are two—so that's not a contradiction, is it? – LSpice Feb 24 '23 at 19:32
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    I quite like mathematical problems motivated by physics. – T. King Feb 24 '23 at 22:12
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    The most important mathematical question is the one you have just asked. – bof Feb 25 '23 at 00:03
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    @LSpice: Sure, many important things are opinion-based. If, though, the important thing under consideration involves judging other people's performance at an institutional level, one will typically expect the opinions to be guided by somekind of intersubjective understanding of what constitutes a good performance. A figure skater might get different scores for the same performance, based on each juror's individual opinion. But this does not render the question "What makes the performance of a figure skater good?" completely opinion-based. – Jochen Glueck Feb 25 '23 at 14:02
  • @JochenGlueck I agree with you that the question "Which kinds of mathematical problems does the current mathematical community find important?" is important, has real-world consequences, and has reasonably objective answers. But that isn't the question that the OP asked, and (in my opinion) the OP's question is indeed opinion-based and off-topic for Math Overflow. I think that the OP's question would be on-topic for Philosophy SE, and your proposed alternative question would be on-topic for Academia SE, but neither question is on-topic for this network. – tparker Feb 25 '23 at 16:01
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    Personally, I think that the level of generality of the question asked is too narrow, i.e. a righter, sic!, question would something like: what is an important question about anything? One could try to answer this that the one that provides more meaning than other questions by its existence or by its solution or ... . But then, how do we get to a meaning? Welcome to the question of existentialism, or a Gromov's ergo-mathematics. :) – P. Grabowski Feb 25 '23 at 16:33
  • Sorry and how is it hard to see that this Question depends far more - perhaps solely - on the Questioner's mental state than on any logic? – Robbie Goodwin Feb 26 '23 at 20:02
  • The one that will end up with an important mathematical answer. – Amir Asghari Feb 27 '23 at 12:42

11 Answers11

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The question what makes a mathematical problem worth studying and even important is itself an important meta question about mathematics. Here are a few points (at time subjective) one can consider

  1. Difficult - For a problem to worth your effort it need to be difficult. If it is easy or the solution is routine, it is a weakness.
  2. Not hopeless - Both for individual mathematicians and for mathematics as a whole, problems that are utterly hopeless, are less worthy.
  3. Deep - I will not try to define depth.
  4. Fundamental - There are problems which are obviously fundamental to a mathematical area like what are the finite simple groups (or infinite simple groups) and what is the structure of p-groups. Sometimes the solution leads to fundamental insights.
  5. Requires new tools
  6. Natural - I tend to see problems that come up naturally as important. (But this is controversial.)
  7. Beautiful - Mathematicians may have different tastes for beauty but individual tastes are not orthogonal.
  8. Connected to other problems; (added by Dirk) helpful to resolve other problems or understand them better.
  9. Applied; connected to other areas of science and technology
Gil Kalai
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    I observe an ever-so-slight shift in wording here: the OP asks about "mathematical questions" and this answer is about "mathematical problems". Problems are questions, but are questions always problems? To me, "problem" sounds more well-defined, and once a problem is solved, it is solved; while a "question" could be very open-ended. (Gowers's "problem solvers" and "theory builders" come to my mind.) – Jukka Kohonen Feb 24 '23 at 14:38
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    Since this is CW, I took the liberty to add "useful" to the list because I think that this is in the spirit of the other points. Feel free the remove it again, though. – Dirk Feb 24 '23 at 15:26
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    Since Gil is a well known figure in mathematics, there will be wide interest in his view on this, so I'm going to roll the edit back. – HJRW Feb 24 '23 at 16:50
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    Why should hopeless questions be not important? -- For example, a hopeless question may still motivate work which yields important insights. – Stefan Kohl Feb 24 '23 at 22:42
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    I think that for a problem to be utterly hopeless (either for an individual or for the community) is a weakness. (Of course this is also a matter of subjective assessment and, as you said, hopeless question can also lead to important insights.) Here is an example: The question if there is always a prime in the interval $[n, Clog^2n]$ seems hopeless and less important than the RH which is extremely difficult but still somewhat in the boundary of what could be expected. – Gil Kalai Feb 25 '23 at 07:38
  • I incorporated Dirk's suggestion in item 8. – Gil Kalai Feb 25 '23 at 19:26
  • @JukkaKohonen Good point. (I think we can replace "problem" by "question" in my answer.) – Gil Kalai Feb 26 '23 at 20:37
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    "Problems worthy of attack prove their worth by fighting back." --Piet Hein – Joseph O'Rourke Mar 01 '23 at 19:20
  • The question/problem distinction is relevant. "What is the explanation of monstrous moonshine" was a question before it was a problem; similarly Grothendieck's hunt for "motives" in algebraic geometry. Idk about hopelessness being relevant even for problems. RH and P vs. NP are presently hopeless problems, but still important. Solving 3n+1 will take entirely new methods, and only after seeing those methods will anyone be able to tell whether the 3n+1 problem is important. – none May 23 '23 at 00:24
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I'd rather ask "how is" a math question important as opposed to "what is" an important mathematical problem. Some examples

  1. (Solves a bottleneck) If the question you address is the first in line of a bunch of related but unaccessible problems, and by giving answer to the first you imply the truth/falsehood of all problems thereafter, this problem is important.

  2. (Develops a tool) If the method you introduce to address the question can be vastly reused, then the question was an important one.

  3. (Captures the essence in a 'common phenomena') If the question you ask leads to a useful definition or useful conceptual framework for capturing the underlying phenomena you are trying to reason about, then you've asked an important mathematical question.

  4. (Fashion / Timing) The importance of a mathematical question can rise and fall like fashion. If you are a Pythagorean, figuring out the diagonal of a square was an important question. If you are a mathematician today, maybe the Langlands program is more important.

I think sometimes an important question is one a lot of people cares about and don't know how to answer, but other times, it might be just a handful of people asking the question, yet from having answered this question you are led to a whole bunch of discoveries.

The key word seems to be impact, and where the question leads in the network of mathematical ideas.

(Edit)

A bit more down to earth, no one is going to wake up each day and ask a question that is going to go down history as one of the top 1% questions ever asked. But if you ask a question that by thinking through you refreshed your understanding or just think time was not wasted then it's an important question for you to ask. If you develop this habit of asking a question that advances your understanding, and one day you happen to be working at the frontier of mathematics where nobody understands things quite so well, the questions you ask may well turn out to be important mathematical questions.

KConrad
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I think most people would agree that the criteria Gil Kalai highlighted are what we, as a community, would say to the layman, or to politicians, or other scientists (say biologists or chemists). But since we are on a math website, I am naïve enough to think we can be more honest.

In most cases, what makes a question important is that the big shots in your field decided that the question is important. This is perfectly illustrated by your feeling about Schur positivity. The big shots in representation theory decided that some questions are important and you feel that people are much more inclined to listen to you if you relate your interesting problem to the questions of the big shots.

But then you may ask "How to define a big shot". The answer is very simple : this is someone who solves important questions. And you get a perfectly virtuous circle of "important people" asking important questions and their students (or post-docs) solving them, becoming in the process the new big shots.

A long (long) time ago, I asked my master thesis adviser why is it that the students of such or such big shot would get permanent position so early in their career, while some other young researchers, who looked more productive, more original and more dedicated to Science in general, would have so much trouble to find a job. And he told me "The answer is very simple : the big shots get the best students to do a Phd with them, they give them the important and interesting questions for their theses and then the students get the best Phd theses, so they get the best positions early on in their career." Crystal clear!

Edit added later : After afew more thoughts, I would like to argue that the word important should be avoided to describe a mathematical question. "Well-motivated, surprising, fun, well-connected,..." seem to me much more adequate to describe the mental reactions you may have when facing a (new) question.

"Important" is in my opinion a word of power that people with sociopathic behaviours in academia (see for instance this paper) may use to break the career of someone not working in their area ("this paper is not worth being published, it deals with unimportant questions") or to favour the careers of those closely working with them (" this Phd thesis is awsome, it deals with a very important question").

Libli
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    As a representation theorist myself, but not a big shot, I do not get to decide unilaterally what questions are important, but I hope I may put in a very strong vote for the abbreviation "rep. thy." over "rep. the.". – LSpice Feb 24 '23 at 19:36
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    @LSpice : okay, i will edit. – Libli Feb 24 '23 at 19:38
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    As a layman, I did not understand what "rep. thy." meant until I reached @LSpice's comment, so I took the liberty to replace this abbreviation with the full "representation theory", such that people like me should understand the text immediately. – Alex M. Feb 25 '23 at 08:50
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    -1: after 30 years in mathematics and very definitely not a "big shot" myself, this does not represent the corner of mathematics I live in, at all. In every area of mathematics I have worked the "big shots" are incredibly talented and hard working, and the problems they consider "important questions" are with very few exceptions rooted in being hard to solve and linked with other stuff in the theory. – Martin Argerami Feb 25 '23 at 15:06
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    @MartinArgerami : if you read my answer with care, you will notice that I wrote "in most cases". Perhaps you are the lucky one who works in an area of mathematics where the big shots are creative, supportive of young researchers (even the ones not working underground their supervision), not judgemental and clear sighted. – Libli Feb 25 '23 at 16:30
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    @MartinArgerami : let's close this discussion with some dignity. I certainly don't want to engage in an argument with you about who has the bigger (list of publications, list of books written, list of committees we took part in,...). This is typically the kind of sociopathic behaviours I highlighted in my edit, and I definitely would like to refrain from behaving in such a toxic way. Anonymity enables me indeed to state some facts, which I could not state with my real name. The point is that the academia is dysfunctional and highly toxic in many ways and we need to think and work on it. – Libli Feb 25 '23 at 17:16
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    @Libli: there is toxicity, as there is in any human endeavour with more than a handful of persons. And I didn't want to imply that my cv is bigger than anybody's which it isn't in many many cases; I just wanted to justify that I believe I have a decent outlook. I still think the picture you give makes it sound as toxicity is prevalent, which I don't agree with. – Martin Argerami Feb 25 '23 at 18:54
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I want to point out that you raised two questions, and in my opinion they are very different questions.

  1. So I really want to know how to decide whether a question is worth studying?
  1. How do I decide what question is important to ask in mathematics?

The answer to Question 1 lies in your personal values, whereas the answer to Question 2 lies in community values.

Importance, as others have explained well, is inherently a question about what the mathematical community values. The way you learn what is important is by studying the words and actions of the mathematical community.

Deciding whether a question is worth studying, however, is a personal choice. It can, of course, be informed by knowledge of what the mathematical community values. But in the end, it's your time, your mind, and your life that you are making a decision about. Only you can make the final determination of what is worth investing yourself in. Part of becoming a mature scholar is setting your own internal compass.

Of course, there are practical realities to consider. If the community is offering you something that you want, and will give it to you only if you work on things that it considers to be important and that you do not consider to be important, then you may choose to compromise. Nevertheless, even in such situations, I think you will make wiser decisions if you clearly distinguish between what you value and what others value.

Timothy Chow
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Gil gave you an excellent answer already. I'll just add a small piece of advice: when you hear that somebody says that a certain open problem/question is worth studying, add 1 point to the problem score on your scoreboard and when you hear that somebody says that a certain open problem/question is not worth studying, subtract one point from the score of the person who said that.

fedja
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    To reduce incivilities, people are very hesitant to say negative things. On the other hand, people regularly boast and exaggerate, especially about things they do. So, the subtraction constant should be far larger than 1. – Boris Bukh Feb 24 '23 at 18:21
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    @BorisBukh Or, perhaps, the addition constant should be far smaller than 1 :-) – Carl-Fredrik Nyberg Brodda Feb 25 '23 at 11:52
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It is indeed somewhat subjective. A discussion of this, with examples, is contained in Hardy's book Mathematician's Apology. But mathematicians frequently disagree on many questions whether they are important or not. For example, Vladimir Arnold disagrees with Hardy in many cases. And Jacobi disagreed with Fourier, in their famous exchange.

Alexandre Eremenko
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Words like "important" and "natural" are obviously subjective, and that's OK! It shouldn't surprise you to learn that other people have opinions about the value of certain mathematical ideas, just as you probably hope that they aren't surprised that you have opinions, too.

That said, your question suggests a comparison between two types of value judgements: "important" or "natural" vs. "fun" or "curious". The first type suggests that people should work on the problem out of some sense of obligation, whereas the latter suggests that it would be pleasurable to do so. It sounds like you're asking: where does this sense of obligation come from?

The answer is once again obvious: it's determined informally by how the mathematical community allocates resources - journal space, grant money, academic jobs, and so forth. The community has certain informal standards, expressed very beautifully by Gil Kalai's list, for instance.

But presumably those standards exist to ensure that the mathematical community is achieving some sort of larger objective, which makes it worthy of investing resources in the first place. I think that objective is something like "organize and preserve mathematical knowledge". As more and more mathematical knowledge is generated, it becomes harder for even a modestly sized group of experts to keep track of it all. So in order for all of that knowledge to survive in a form that can be easily consumed and appropriated by future generations, there has to be a constant process of simplifying and clarifying the most fundamental ideas.

So that's what I think "important" means. A student today can graduate college knowing how to solve a dozen problems each of which past mathematicians spent their entire lives working on, and it's because the intervening generations isolated a small number of crucial ideas that tie it all together - things like the definition of a limit, or Fourier series, or Galois theory. I think many mathematicians today feel an obligation to provide a similar service for students of the future.

Paul Siegel
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As Gil Kalai mentions in his answer that he "will not try to define depth", here's a possible complement to his answer. John Stillwell has an excellent lecture on this question and its history, available here: "What Does 'Depth' Mean in Mathematics?".

One particularly nice example I enjoy from there is early on, where there are four commonly-accepted-as-deep theorems presented: Dirichlet's theorem on primes in arithmetic progressions, the Poincaré conjecture, Fermat's last theorem, and the Classification of Finite Simple Groups. One overarching reason he mentions that, even though they are all theorems about discrete objects, one of the things that makes them deep is that somehow continuity enters into the proofs. Furthermore, he goes on to use history as a gauge of depth, and mentions (as a high-level, summary, idea) that:

[the theorems are deep because] it took a long time to prove them, they involves several stages, and generally it was very good mathematicians who worked on them.

He then gives the excellent definition of depth as "the number of shoulders of giants that one must stand on to reach the result"!

I recommend anyone interested in this topic to watch the full lecture.

Carl-Fredrik Nyberg Brodda
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Beauty. Beauty is important factor. A question may appear irrelevant at first, but if it admits a beautiful proof, that may make the result important. And it does not need to be complex, a proof may be dazzling even in its simplicity.

Andrea Alciato
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Today the iso 2023-02-24, according to mathoverflow, here are the important questions in mathematics (vote > 200, the word important appearing somewhere).

What to do? is the most important question.

Why is a topology made up of open sets ?

Examples of common false beliefs are also important.

Mistakes are important.

The axiom of choice.

Knowing what is the best algebraic textbook. For some reason, Hartshorne is disqualified.

Again why is a topology made up of open sets ?

The surprising connections in mathematics are important.

What makes dependent type theory more suitable than set theory for proof assistants?

What are the fundamental examples is important.

Philosophy behind Mochizuki's work on the ABC conjecture.

Widely accepted mathematical results that were later shown to be wrong are important

Finally, rigour is important.

Other questions have not received much votes or failed to name themselves as important, so they can be decently ignored.

coudy
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In my experience, often "importance" flows from big, challenging problems down to esoteric, slightly-less-challenging problems, in a long convoluted stream that may become totally obscure.

In computer science, we care about P vs NP. Well, we can't solve that. What would be a step toward solving that? Maybe showing that 3SAT requires superlinear time or space. Well, we can't show that. Maybe we can show that restricted classes of circuits can't solve 3SAT (or an even simpler problem). Okay, if that's hard, maybe we can formulate an algebraic version of the question. Okay, solving that involves understanding how intricate a low-degree polynomial can be. Etc.

Oh, but maybe we can't solve that problem either, but we can formulate an analogous problem for polynomials of certain degree over a different finite field, and hope that techniques and ideas developed could lead to breakthroughs in $\mathbf{F}_2$, or whatever (I am not an expert in the area, so I'm making things up at this point).

In the end, you have a continuous stream of "important" papers with results such as improving bounds on properties of the Fourier spectrum of certain polynomials, and newcomers (perhaps even oldcomers) can't trace the motivation any more.

usul
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    The OP’s question is about items of genuine importance, either to the OP or to a wide community. So for items whose importance is only so-called, the comments here say something sensible, but don’t provide much of an answer to the question. –  Feb 25 '23 at 20:06
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    @MattF. maybe I didn't communicate well or ended on the wrong note. A takeaway is that seemingly obscure problems can be genuinely important by fitting into a grander program in this way. – usul Feb 26 '23 at 04:09
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    You could convey that message more clearly by giving a real example of an obscure problem and then arguing for its importance in this way. Do you have an example of a paper proving a theorem for an apparently unimportant field, and a later paper adapting it for a more obvious field? –  Feb 26 '23 at 10:14
  • @MattF. I'm not sure why the "later paper adapting" part would be needed to support my point, but just looking for examples along the lines of my narrative on P/NP, the most recent FOCS proceedings contains this which uses invariant theory to construct pseudorandom generators for polynomials of small degree over large enough fields https://www.computer.org/csdl/proceedings-article/focs/2022/551900a399/1JtvKspTWQE or the most recent STOC proceedings contains similar examples e.g. on constraint satisfaction over a cyclic group in a streaming setting https://dl.acm.org/doi/10.1145/3519935.3519983 – usul Mar 08 '23 at 02:32
  • The post suggests that, as an attack on proving something over the important field $\mathbf{F}2$, one might prove the same theorem over $\mathbf{F}{27}$ first. Neither of the cases in the comment is an example of that pattern. –  Mar 08 '23 at 10:05
  • @MattF. hence "along the lines", although the first paper is indeed presenting a result that lowers the necessary size of the field significantly compared to prior work. If you would like more explanation on why these are obscure problems but are indirectly connected to large programs such as P vs NP or P vs BPP, I would be happy to continue the conversation in chat. – usul Mar 08 '23 at 18:20