I think the issues you describe are not as closely connected as they seem. I think that, by the standards of pure mathematicians, my research interests are unusually broad (certainly if measured by the topics I think about enough to answer mathoverflow questions about).

However, I don't think I would have any special ability to avoid a situation where some result I am working on has already been proved. Everything I learn about the current state of research in one area comes at the cost of learning less about the current state of research in another area. The fundamental problem is the increase in the total amount of mathematics that is produced. You can only keep track of so much of this, and however you divide it up, whether narrow and deep or shallow and broad, you are likely to miss things.

The easiest solution is to work in a narrow area where you can be confident you understand pretty much everything that is done. Indeed, I think this problem is part of the motivation for mathematical specialization, rather than a consequence of it.

I also have begun skipping seminars in my main topic areas. Part of the reason for this is that at my current institution there are three seminars in these areas, while when I went to grad school there were only two. I rarely go to seminars in my areas of secondary interest but if I did I would probably go to seminars in my primary topic areas less.

I don't know, but suspect that there is an overall trend to having more seminars, conferences, and things of that nature, causing people to be more selective about which ones they go to. Surely the post-COVID explosion in online seminars has had some effect in this direction.

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  $\begingroup$"The fundamental problem is the increase in the total amount of mathematics that is produced." Amen to that. Something is fundamentally off when the typical research paper has something like 0.01 readers on average (my estimate, not based on hard data, but I don't think I'm underestimating the number of readers here). This is the basic problem, and much of what the OP describes (every moderately interesting theorem is proved and published several times by different people etc.) are inevitable consequences.$\endgroup$

  Christian Remling

  – Christian Remling

  2026-02-16 17:00:27 +00:00

  Commented12 hours ago
• 2
  $\begingroup$I certainly agree with this. Mathematics I believe actually does not suffer as badly as other scientific disciplines, where asingle journal can publish upwards of 10,000 papers a year! How's anyone supposed to absorb all of that? Of course, even in mathematics the quantity of insipid papers is still staggering.$\endgroup$

  Stanley Yao Xiao

  – Stanley Yao Xiao♦

  2026-02-16 17:08:32 +00:00

  Commented12 hours ago
• 3
  $\begingroup$@ChristianRemling How does this .01 readers thing work? Since the average mathematician writes less than 100 papers in their career, doesn't that mean the average mathematician would have to read less than 1 paper in their entire career for the .01 readers/paper claim to work out? I don't think that is accurate.$\endgroup$

  Will Sawin

  – Will Sawin

  2026-02-16 17:25:25 +00:00

  Commented12 hours ago
• 2
  $\begingroup$In the areas I follow I think that important papers get read carefully by many people and most papers good enough to get published by reasonable journals get read carefully by at least a couple of people. I read at least the introduction of every paper I become aware of that touches on topics I actively work in. Of course, there are plenty of papers that are really bad, but I don't think that it's such a big deal that they are typically read by <=1 person.$\endgroup$

  Andy Putman

  – Andy Putman

  2026-02-16 18:13:28 +00:00

  Commented11 hours ago
• 1
  $\begingroup$@ChristianRemling I think even the modified claim is a bit too pessimistic. Generally bibliometrics show that over 50% of papers have at least one citation. It's possible to cite a paper without reading it, but I think for the lower end of papers, it's more likely someone reads it without citing it than cites it without reading it. (How often will someone read such a paper, realize it's useless for what they are trying to do, and give up without citing?) So I would say a typical paper is read 1-5 times beyond referees, maybe.$\endgroup$

  Will Sawin

  – Will Sawin

  2026-02-16 19:14:58 +00:00

  Commented10 hours ago

In Strasbourg, we have a regular very well-attended seminar we call "sem in" where local people give introductory level talks about something from their corner of mathematics that should be accessible to working mathematicians of all disciplines. This way we keep awareness of what colleagues in far away disciplines do, and also figure out how to explain at least some aspects of our own specialties in a very elementary way. If this (and not showcasing your best theorems) is the declared goal of a seminar, that certainly helps achieving the goals you mention.

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  $\begingroup$The math departments of Harvard and MIT had occasional joint seminars, where the speaker (local faculty or visitors) could talk about any mathematics they wanted, subject to one rule: You're not allowed to mention any of your own theorems. These seminars were aimed at a general mathematical audience, and I (a grad student at the time) found them well worth attending.$\endgroup$

  Andreas Blass

  – Andreas Blass

  2026-02-16 20:39:06 +00:00

  Commented9 hours ago
• 2
  $\begingroup$I have also heard seminars like this called “basic notions” seminars.$\endgroup$

  Sam Hopkins

  – Sam Hopkins♦

  2026-02-16 21:03:09 +00:00

  Commented8 hours ago
• $\begingroup$@SamHopkins it does not really have to be a basic notion to merit a down-to-earth explanation though! I was really impressed by pedagogical finds of some colleagues who do highly abstract things daily but made an effort of explain it in a beautifully simple way at the sem in.$\endgroup$

  Vladimir Dotsenko

  – Vladimir Dotsenko

  2026-02-16 21:05:25 +00:00

  Commented8 hours ago
• $\begingroup$This reminded me of the AMS "What Is..." series.$\endgroup$

  Robin Saunders

  – Robin Saunders

  2026-02-16 21:31:23 +00:00

  Commented8 hours ago
• $\begingroup$We have a similar seminar in our department. First, attendance was great with full room, including many faculty. After few years, only few people come, almost all graduate students. Sigh.....$\endgroup$

  Moishe Kohan

  – Moishe Kohan

  2026-02-17 05:18:58 +00:00

  Commented38 mins ago

One thing that has helped in my experience—both personally and institutionally—is the presence of problems, phenomena, or applications that are intelligible to a broad audience. I think most people (even mathematicians) appreciate concrete examples. When I try to tell people about a new statistical approach, it helps to have a concrete example of a dataset where existing approaches fall short. When I tell people I study topology, it helps to have concrete examples of topological spaces people are interested in, like the shape of the universe, or shapes in topological data analysis.

Interdisciplinary centers are great ways to incentivize breadth. For example, at places like theMathematical Biosciences Institute at Ohio State University, mathematicians working in very different subfields routinely interact because biological questions provide a shared language. A topologist, a probabilist, and a PDE analyst may not share techniques, but they can all understand why modeling immune dynamics or ecological systems is interesting. The application acts as a conceptual meeting ground, and encourages researchers from all these areas to understand better what the other ones are doing.

Similarly, areas such astopological data analysis have drawn participants from topology, statistics, computer science, and data providers like neuroscience, around a common reference point. The data situations encourage topologists to learn broadly (e.g., machine learning algorithms) and also to push the boundaries of topology more widely rather than deeper. It turns out there were interesting research questions accessible to a first-year graduate student, e.g., faster algorithms for computing homology using sparse matrices, but in order to discover those questions, it was important to stop and ask what a general audience could need from topology, and how we could use our expertise to help them with the research questions they care about.

You can see the same phenomenon institutionally at theBeijing Institute of Mathematical Sciences and Applications (BIMSA), which frames many pure mathematics activities around themes such as understanding the mathematical underpinnings of artificial intelligence and why LLMs act the way they do. In addition to being of interest to a wider set of people, this kind of thing can also help spur new research at a level that's more comprehensible to a larger audience, e.g., new developments related to dynamical systems on graphs, rather than theory for the sake of theory that will only be read by ten people.

I don't want to give the impression that pure math is useless or that I think applied work is inherently more valuable. I've done plenty of "theory for the sake of theory" research as well as a few applied things. But now, because of that applied work, even when I do highly theoretical and deep work, I can explain it to a broader set of people using analogies, metaphors, and potential applications to things they care about. I used to think it was "dirty" to try to "sell" my pure work as applicable in some way, e.g., for the broader impacts part of a research grant. But as I matured as a researcher and teacher, I started to understand the value of broader impacts and a broadly comprehensible motivation, even if the goal is continuing to develop pure mathematics.

My view now is that when a line of research can be connected to a broader mathematical narrative or to questions recognizable outside the subfield, that connection acts as a translation layer. It lowers the cost for others to engage, encourages interdisciplinarity, and ends up making our work have a larger impact to improve the world. So, in addition to research centers like the examples above, we can "incentivize breadth" by cultivating habits of articulation in both seminars and papers around motivating things to a broad audience, encouraging and rewarding expository writing, and broadening what we "count" as scholarship in hiring, tenure, and promotion, to reduce barriers to folks who want to broaden themselves but don't want a negative career consequence when the new work they begin to do is at a lower level for a while (because they are learning the new area) compared with their PhD training. If your department decides what good scholarship looks like, then this kind of decision is actually in your control.

One approach that has been proposed to counter the fragmentation of mathematics is to use its history as a unifier:

History has gained new appeal as mathematics fragmented into morespecialized subfields. Specialized research mathematics can often onlybe understood by ones closest colleagues. History, by contrast,provides a rich store of mathematical material that is accessible andinteresting to all mathematicians. Therefore one of the advantages inthe study of the history of mathematics is to bring colleaguestogether around shared experiences and interests.

Historiography of Mathematics from the Mathematician’s Point of View, V. Blasjo (2021).

I’m not sure how to address the problem in general, but concerning this quote:

Ideally, any two mathematicians should be able to understand why the other does what they do and their motivation for doing so.

I think this is a very reasonable goal, and it is not too difficult in practice. From personal experience, I have little more than a passing familiarity with first year graduate algebra. (if even that…)

But even with this many of my algebraically inclined friends have been able to explain their work and motivation in substantial detail, by relating it to concepts I do know, making metaphors that may not be precise but are meaningful to me. Likewise, I’ve had little trouble explaining my work in analysis to people that have only basic exposure to analysis but have a broad general background.

I think broad exposure to mathematical subjects, even if basic at the level of a first course drastically improves communication possibilities. A lot opens up once you have multiple points of entry into the subject.

Here is a quote from Israel Gelfand, from "Mathematics as an Adequate Language" in "The Unity of Mathematics: In Honor of the Ninetieth Birthday of I.M. Gelfand" -

This conference is called “The Unity of Mathematics.’’ I would like to make a few remarks on this wonderful theme.

I do not consider myself a prophet. I am simply a student. All my life I have been learning from great mathematicians such as Euler and Gauss, from my older and younger colleagues, from my friends and collaborators, and most importantly from my students. This is my way to continue working.

Many people consider mathematics to be a boring and formal science. However, any really good work in mathematics always has in it: beauty, simplicity, exactness, and crazy ideas. This is a strange combination. I understood earlier that this combination is essential in classical music and poetry, for example. But it is also typical in mathematics. Perhaps it is not by chance that many mathematicians enjoy serious music.

This combination of beauty, simplicity, exactness, and crazy ideas is, I think, common to both mathematics and music. When we think about music, we do not divine it into specific areas as we often do in mathematics. If we ask a composer what is his profession, he will answer, “I am a composer.’’ He is unlikely to answer, “I am a composer of quartets.’’ Maybe this is the reason why, when I am asked what kind of mathematics I do, I just answer, “I am a mathematician.’’

As somebody who personally tends to value mathematical breadth a lot, let me play devil’s advocate. Most of the following points admit counterpoints, but deserve to be said:

• Breadth can come at the expense of depth or precision. (and aren’t depth and precision the soul of mathematics?)

• As a student, it’s your job to learn something deeply. It’s your advisor’s job to supply to context. You have to learn to be deep before you can become broad. (And as an advisor: your student might need to focus on depth right now rather than breadth.)

• Breadth can make you vulnerable to groupthink and dull your originality.

• Breadth itself can be a specialty. If you have a broad collaborator, you don’t need to be broad. Heck, chatgpt has pretty broad knowledge of the literature. Why not specialize in your own thing?

• Personally, I was drawn to my own specialty of category theorybecause it seemed like a “shortcut” to acquiring mathematical breadth. But not every mathematician is (or should be) a category theorist.

• Personally, I was drawn to places like MathOverflow because they were a place to be exposed to a wide variety of math. But not every mathematician is (or should be) a person who hangs out on the math internet.

• Category theorists and logicians are both prone to engaging in metamathematics. They’ll say “Ordinary mathematicians are interested in X - what can we say about it?” as though studying an alien species or conducting a sociological experiment. Personally I often enjoy thinking this sort of parlor game. But it would be terrible if all mathematicians were just doing this all the time!

You probably can't change the community, but you can change yourself; to your own, and to the community's benefit.

If you are an early-career researcher (and perhaps later on too) look for opportunities to attend high-quality colloquium-style talks, locally or on-line, and read such texts. This will broaden your knowledge, enabling you to understand the work of a larger percentage of mathematicians. Importantly, it will also help you to develop a better feeling of what types of questions other mathematicians of our time most care about. Therefore, you will be better able to explain your work to others, and moreover, you will become better at choosing your projects so as to maximize your potential audience.

My own career benefited greatly by attending colloquia on every possible topic. I would often only get an impressionistic feeling of the results/proofs, but they were an enriching experience in terms of appreciating the breadths and depths of mathematics, the current trends, and -importantly- their motivations. The best talks were those that offered a glimpse into something profound. The worst were those that explained something basic at undergraduate/PhD level, leaving no time to get into real substance. Much better than having someone explain to youwhat they are doing is to have them explainwhy; and it should be something substantial enough to make it worth it.

• sociology-of-math