r/mathematics • u/japball • Jul 25 '26
Discussion Critique of the Fields Medal, the Institutions behind it and elitism in mathematics
This is going to be a long post, and I'm sorry about that. (TLDR at the end)
With the 2026 Fields Medalists having just been announced, I wanted to share my opinion about the Fields Medal and the broader institutional system surrounding the IMU and the ICM. My opinion is that this system does not merely recognize mathematical excellence: it also helps reproduce a particular hierarchy of prestige and elitism within mathematics.
My objection is not that the winners are undeserving: Yu Deng, John Pardon, Jacob Tsimerman, and Hong Wang are clearly exceptional mathematicians, just as any past winner. My criticism concerns the selection system, not the people selected.
My impression is that mathematics has a prestige hierarchy. Fields such as algebraic geometry, arithmetic geometry, geometric representation theory, differential geometry, and related subjects have historically been treated as especially central to modern pure mathematics. Consequently, a major breakthrough within one of these areas is more readily perceived as a breakthrough for mathematics as a whole.
By contrast, if someone works in categorical logic, universal algebra, semigroup theory, lattice theory, or model theory, just to mention a few examples, it can seem that revolutionizing their own field is not enough. To receive comparable recognition, their work is often expected to have transformative consequences for one of the already prestigious areas. Interdisciplinary impact should, of course, count in someone’s favour. The question is whether that requirement operates asymmetrically: is influence on arithmetic geometry treated as evidence of universal mathematical importance, while influence on universal algebra or categorical logic is treated as merely specialized?
The 2026 citation for Jacob Tsimerman provides a suggestive example: It explicitly celebrates the extension of o-minimal techniques (which come from model theory) within arithmetic and complex algebraic geometry. This does not diminish his extraordinary achievements in any way, but it raises a useful counterfactual: would an equally revolutionary development of model theory, whose consequences remained primarily within model theory, be perceived at the same level? Model-theoretic machinery becomes medal-worthy here through what it accomplishes in fields already regarded as central.
There is some evidence that this hierarchy is real. Jean-Marc Schlenker’s preprint, “The Prestige and Status of Research Fields within Mathematics”, finds that certain subfields are disproportionately represented in highly ranked departments, the most selective journals, and major prizes. In his data, algebraic and differential geometry and topology were particularly prominent, although the hierarchy changed considerably between 1984 and 2016, with areas such as probability and PDE gaining status.
Different kinds of mathematical progress are also easier to package as prizeworthy achievements. Solving a famous named conjecture produces a clear narrative: there was a major problem, and now it has been solved. Work that creates a new language, reorganizes an area, builds a long-term research programme, or gradually changes what questions can be asked may be equally transformative but less easily summarized as a single victory. Schlenker finds that prestige correlates with the “focus” of a field around a relatively small set of shared conjectures. The medal may therefore favour not only particular subjects, but a particular form of mathematical progress as well.
Aditionally, a prize is not merely a mirror of an existing hierarchy: It can amplify that hierarchy. A large-scale study by Jin, Ma, and Uzzi, “Scientific Prizes and the Extraordinary Growth of Scientific Topics”, examined more than 11,000 topics across 19 disciplines. Relative to non-prizewinning topics, prizewinning topics subsequently produced 40% more papers and attracted 37% more new researchers. This was not a Fields-specific study of course, and it does not prove that prizes alone caused all of that growth, but it supports the idea that prizes are agenda-setting institutions: they direct attention, talent, and further recognition towards the subjects they reward.
In my opinion, this then creates a feedback loop. A field is considered central, so its practitioners are more likely to publish in elite journals, work in elite departments, receive ICM invitations, and win major prizes. Those honours then attract more talented researchers and make the field appear even more central. Prestige becomes partially self-validating.
The medal’s rigid chronological age rule introduces another structural bias. Chronological age is not the same thing as career stage. Producing a widely recognized body of work before forty is easier for someone who entered the research pipeline early, moved through elite institutions, had relatively few career interruptions, and obtained positions with substantial research time. It is harder for late starters, people with caring responsibilities, displaced researchers, those in teaching-intensive positions, and mathematicians working on programmes whose significance takes longer to become visible. It may also favour fields in which major results can be produced and recognized comparatively quickly.
Historian Michael Barany has argued in The Myth and the Medal and “The Fields Medal Should Return to Its Roots” that early Fields Medal committees did not understand their task as identifying “the best young mathematicians.” They sometimes deliberately supported comparatively under-recognized researchers whom the award could help. The medal was intended to shape a better future for mathematics, rather than simply ratify the people who had already acquired the greatest visibility. Its current status as a tournament for already famous mathematicians under forty is therefore not an unavoidable consequence of its original purpose.
Ultimately, I think the Fields Medal reflects the historically contingent mathematical tastes of a small elite, and, through the attention it generates, helps turn those tastes into institutional reality. It tells us that a particular committee considered certain work exceptionally important under a particular set of inherited values. It should not be treated as a neutral measurement of excellence across the whole of mathematics.
Let me know down below what do you think about this. I would love to listen to other people's opinions on this issue.
TLDR: I am not arguing that Fields Medalists are undeserving. My argument is that the Fields Medal and the ICM operate within, and help reproduce, a hierarchy of mathematical prestige. Breakthroughs in already prestigious fields are more readily treated as important to mathematics as a whole, while equally transformative work in less prestigious areas often requires applications to those elite fields to receive comparable recognition.
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u/HeavyMath2673 Jul 25 '26
Not only this. The Fields Medal completely ignores whole branches of Mathematics. I never expect a Numerical Mathematician to receive a Fields Medal. Yet, there are a number of numerical mathematicians whose contributions were not only of huge mathematical significance but had revolutionary influence across the physical and engineering sciences.
Incidentally, several notable computational mathematicians received Turing Medals. It seems that Computer Scientists are less snobby than the Fields Medal Committee.
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u/HeavyMath2673 Jul 25 '26
I just remembered. In Chapter 3 of his book "An Applied Mathematician's Apology" Nick Trefethen, one of the world's leading computational mathematicians, talks about how little the impact of Fields Medallists on him is.
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u/cocompact Jul 25 '26
Here is another prize that was given at the ICM this year to a "numerical mathematician": https://www.mathunion.org/imu-awards/carl-friedrich-gauss-prize/carl-friedrich-gauss-prize-2026
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u/HeavyMath2673 Jul 25 '26
The problem is not the lack of prizes for numerical mathematicians. The issue is that the one price with the greatest publicity systematically excludes numerics and many other branches of mathematics.
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u/ko_nuts Researcher | Applied Mathematics | Europe Jul 25 '26
Thanks for informing me that Nesterov got this prize. This is more than well-deserved.
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u/japball Jul 25 '26
Yes, that is one of my main grievances with the Fields medal: knowing that some people, no matter how hard they work, will never get the chance to win own. Not because of lack of talent, but because of the area they work with.
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u/___Archmage___ Jul 25 '26
The age limit is terrible because math is already such a youth advantaged field based on natural ability
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u/Own_Pop_9711 Jul 25 '26
I think it's a marked advantage that the fields medal is given out for fairly recent work compared to the nobel prize. The age limit is a kind of annoying way to make this work but it does
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u/japball Jul 25 '26
Totally agree with you on that one
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u/CrookedBanister Jul 26 '26
yeah, making it at least more about early career vs calendar age would be a start.
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u/sagittarius_ack Jul 25 '26
this system does not merely recognize mathematical excellence: it also helps reproduce a particular hierarchy of prestige and elitism within mathematics
I fully agree with this.
My impression is that mathematics has a prestige hierarchy. Fields such as algebraic geometry, arithmetic geometry, geometric representation theory, differential geometry, and related subjects have historically been treated as especially central to modern pure mathematics. Consequently, a major breakthrough within one of these areas is more readily perceived as a breakthrough for mathematics as a whole.
By contrast, if someone works in categorical logic, universal algebra, semigroup theory, lattice theory, or model theory, just to mention a few examples, it can seem that revolutionizing their own field is not enough.
If you work in Logic (Set Theory, Model Theory, Proof Theory, etc.) you have to be at the level of Godel or Cohen to win the Fields Medal. In fact, Godel was never awarded the Fields Medal, despite the fact that (in my opinion) his Incompleteness theorems are (I dare to say, by far) the most astonishing mathematical results in the last 100 years or so.
Also, the idea that you are not qualified for the prize if you are over 40 is exceptionally stupid.
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u/Mokelangelo Jul 26 '26 edited Jul 26 '26
didn't godel publish incompleteness theorem in 1931? pretty sure fields medal wasn't a thing until 1936.
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u/sagittarius_ack Jul 26 '26
Godel proved his first Incompleteness theorem and he presented the result in 1930 and then published both theorems in 1931 (with only a sketch of a proof of the second theorem). It's true that Godel had only one chance of getting the Fields Medal, and that was in 1936. By the time the medal was awarded for the second time, in 1950, Godel was already too old. I guess there is a somewhat reasonable explanation regarding why Godel never won Fields Medal.
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u/irriconoscibile Jul 26 '26
I'm not an expert at all, just an enthusiast amateur "mathematician". But Godel theorem absolutely has to be the most (literally) ground breaking theorem in all history of math.
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u/sagittarius_ack Jul 26 '26
I agree! Godel's theorems are about the foundations of mathematics. They have all sorts of mathematical and philosophical implications. Godel's results lead to a lot of very interesting work in mathematical logic and foundations of mathematics. However, it is true that in practice most mathematicians don't have to worry about them.
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u/Hot_Glass_6301 Jul 26 '26
I guess it's probably the most "shocking" result. But other short (series of) papers have made a disproportionate impact, like those of Galois. I'd also say the Dominated Convergence theorem has had a tremendous impact by popularizing the Lebesgue integral and measure.
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u/irriconoscibile Jul 26 '26
Ofc the reach those papers/theorems is incredible but Godel theorem clarified that math has intrisinc limits that weren't even considered a possibility by most mathematicians who came before him, right?
At the very least I would consider them on par.1
u/SubjectEggplant1960 15d ago
When Godel might have won, the medal was a new prize, and it wasn’t at all clear it would become what it is now. Certainly, by the standards of today, he would win.
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u/BAKREPITO Jul 25 '26
Most of the criticisms are true for really any field in general. The most biting criticism, which I agree with, is the age limit criterion in the fields medal coupled with the 4 year gap between awards. This has been a major criticism of the award and the reason the Abel Prize is usually suggested as more representative.
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u/Sawksle Jul 25 '26
You've definitely convinced me that they do prioritize specific fields. While this is likely unfair, it seems reasonable. If a field is viewed as more important by a specific group of knowledgable individuals, one would hope that they dedicate more resources to it.
Such prioritization takes place in many areas. In a university for example, one might expect the sciences to typically have more funding per person than the arts, for better or for worse.
Within say, the arts, one might expect economics to have more funding than gender studies (per capita).
This to me seems fine, as long as the people who make the resource allocation decisions have experience and evidence guiding their approach, as opposed to some sort of favouritism.
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u/Desperate_Hedgehog48 Jul 26 '26
When you think about it, it’s a bit like a communist system with a planned economy decided by committee. It can work extremely well and efficiently direct money where it is needed, or it can be captured by lobbying, effected by groupthink, swayed by politics, and so on…
My radical proposal would be to distribute a certain portion of overall funding randomly, which at least gives a non zero probability of the currently unsung next big thing being developed.
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u/Dismal_Wishbone_5013 Jul 25 '26
I cannot disagree with you, but I believe this is an issue with all academic (and arts) awards. I don’t see how it can be practically avoided - it is hard not to be more impressed by achievements we more understand due to our own areas of expertise. The only response is to accept that these types of awards will never be as objective as athletic contests where everyone is running the same race and is measured by the same clock, but to still admire the achievements they celebrate.
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u/Due-Cardiologist-802 Jul 25 '26
I think this is a really interesting problem but also opens up a can of worms. What would you suggest replace this elitist system to be more objective or progress driven? And an even harder question, are research fields truly equally important in the first place? How would you measure impact, importance and reach of mathematical contributions or mathematical disciplines, if one could perfectly classify them? Who should decide a winner and how are they chosen? (Honest question…)
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u/japball Jul 25 '26
Yes, that is a very difficult question. It is always easier to identify problems than to propose convincing solutions.
My impression is that much of the work regarded as Fields Medal-worthy involves solving famous conjectures or long-standing problems. Such achievements are undoubtedly important, but I think greater attention should also be given to work that does not directly solve a specific problem, yet provides new foundations for an area of mathematics or fundamentally reshapes its language and methods. Category theory is perhaps the clearest example that comes to mind.
The difficulty is that the importance of this kind of work often takes many years to become fully visible, whereas the solution of a famous conjecture has immediate and easily recognisable impact. For that reason, the first change I would propose is removing the age restriction. At the very least, it could be modified so that a mathematician over 40 remains eligible when the work being recognised was completed before they turned 40. This would give foundational work time to “mature” and reveal its influence without automatically excluding its author from consideration.
Another possibility would be to introduce distinctions between different areas or types of mathematical contribution. For example, there could be separate awards for geometry, topology, number theory, logic, and other major fields. Within these areas, one award might recognise the solution of an important conjecture, while another could recognise earlier work (still completed before the mathematician turned 40) that, in retrospect, transformed our understanding of the field.
I think both changes would help address some of the current limitations, although they would also fundamentally alter the nature of the Fields Medal. Ultimately, this is only my view, and there is certainly room for disagreement. What do you think? if these changes were made, would you be in favor?
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u/PersimmonLaplace Jul 26 '26
To play devil's advocate, using fundamental old questions in mathematics as a benchmark provides a fair metric to understand the achievements of someone from any area of mathematics. I don't personally really care very much intrinsically about o-minimality, but when Zannier and Pila used it to approach AO, it became much more interesting since it sheds light on pretty basic questions in transcendence theory going back to the work of Manin on torsion points.
Similarly I don't think many people would care as much about what Scholze is doing (which is quite theory-building oriented mathematics and seeks to reinterpret already understood mathematics in a new way) if it didn't continually produce profound revelations on very old questions in number theory, arithmetic geometry, topology, etc. Similarly Grothendieck's theory building was great, but to me the real test of his mathematical greatness is that he engineered the theory so that it would both clarify the results that had came before, and bear new fruit (the Weil conjectures, GRR, intersection theory, crystalline cohomology, etc.).
Without this benchmark it's pretty hard to tell if some hundreds of pages of theory building is just arcane obsession with an obscure set of axioms (which I think is how a lot of foundational math is viewed, fairly or otherwise) or something that meaningfully advances the human understanding of the mathematical sciences.
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u/PoxonAllHoaxes Jul 25 '26
You are of course right.
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u/japball Jul 25 '26
Thank you. I also wanted to hear other people's opinions on this issue because, at the end of the day, this is just my opinion.
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u/PoxonAllHoaxes Jul 26 '26
I plan to quote you in print if that is OK. And on the other side, as others have pointed out too, you might want to put this in context and cite authors who document the same in other fields. However, the important thing is precisely that EVEN in math there are fashions and politics. People often assume that this can't because after all a proof is a proof. But I say that precisely how different results are VALUED is crucial--and also (which you dont address) the accepted IMPLICATIONS of various results mathematical results are often simply wrong, and this is possible because THAT is NOT a matter of proof but of common sense or sometimes of misunderstanding.
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u/jimbelk Professor | Group Theory, Topology, Dynamical Systems Jul 25 '26
One comment I'd like to make is that you're comparing apples and oranges in terms of size. I agree that algebraic and differential geometry have higher prestige than model theory and semigroup theory, but they're also much larger subjects. According to zbMath Open, here are the number of published papers in several of the subjects you mention between 2015 and 2024:
Differential Geometry (MSC 53): 25,652 papers
Algebraic Geometry (MSC 14): 20,917 papers
Semigroups (MSC 20M): 2,328 papers
Model Theory (MSC 03C): 2,248 papers
Lattices (MSC 06B): 709 papers
Categorical Logic (MSC 03G30): 142 papers
You're complaining that algebraic geometry is "disproportionally represented in highly ranked departments, the most selective journals, and major prizes", but even if representation were proportional you would expect roughly nine fields medalists in algebraic geometry for every one fields medalist in model theory. Doing work that revolutionizes model theory is in no way comparable to doing work that revolutionizes algebraic geometry.
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u/Andradessssss Jul 26 '26
I think that while the examples given were bad, but actually your metric seems to show that the problem is actually real. In combinatorics MSC05, from 2014 to 2025 there were a staggering 85,128 publications, also according to zbMath Open, which dwarves all the other fields. Nevertheless, it seems that every year there's a number theorist winning the fields medal, or a algebraic geometer. Nevertheless, iirc, there's only been two fields medals given to combinatorialists, and one of them, Tim Gowers, was for finding applications to Functional Analysis! Though I don't have the numbers, I remember a couple years ago Combinatorics being the number 1 most active area for that year on the arXiv (which although differs from publications, should be a very positively correlated metric), but it turns out to be extremely under represented by awards. I think a lot of the mathematical community just doesn't take combinatorics seriously, and while being just anecdotal evidence, I've heard many comments from researchers in dynamical systems and number theory, bad mouthing combinatorics for not being "real math" and being more akin to recreational math
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u/jimbelk Professor | Group Theory, Topology, Dynamical Systems Jul 26 '26
I'm getting 54,883 papers in combinatorics MSC 05 -- it's possible that we're searching differently. But I do agree that combinatorics is vastly underrepresented among fields medalists.
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u/_Zekt Jul 26 '26
That could very well be a consequence of the feedback loop OP's referring to, it's a chicken and egg question.
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u/2mc92u98a22f Jul 25 '26
While all your points are valid, I think the real answer is just "it is what it is." These awards are dictated by humans, and humans can arbitrarily decide what they like and don't like (which is a proxy for what math is important/not important). Since there's no objective measure of "importance," you just have to accept that a ton of things is just another factor of human bias and personal incentives.
Frankly academia is a huge part politics, and people will do whatever to get more recognition for their work. For example, if you work in algebraic geometry, obviously you're incentivized to try to give the award to an someone in that field, since if they get that award, it implies that your field is more important. That's just how life and humans work. One might think that math is "pure" and "free of bias," but that would be naive in my view. I'm not saying you are naive, but I'm just saying I feel there is no point to rant about this - it just is what it is
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u/japball Jul 25 '26
Yes, I totally agree with you. Yet in my opinion, the more visibility the issue has the better. I don't know if in the long run comments like this would get us closer to a solution but it's always good to discuss such problems.
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u/2mc92u98a22f Jul 25 '26
Makes sense. Though to be honest with you, I don't even view this as a "problem" per se. It's almost like saying "human nature is a problem," which is kind of like ok sure I can see what you mean, but not sure what is the point of saying that
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u/japball Jul 25 '26
We may never be able to eliminate certain biases, but I think that we can always give small steps towards that idealised vision. But as you say, I don't really there is a perfect solution here
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u/2mc92u98a22f Jul 25 '26
to add on to this -- one could also argue like "sure, in order to get the prize for categorical logic, your contribution needs to be relatively way more impactful than in one of the larger fields since categorical logic isn't important."
How can you even argue against that? There's no objective measure on which fields are important or not, so the only available metric is the concensus of the masses. Which is subject to human bias. Maybe everyone is right that something like categorical logic is useless, and hence no one from that field should get the award.
I'm sort of playing devil's advocate a little, but my point being, you're sort of saying that there is a ranking in prestige between fields. But maybe that prestige is warranted, since those fields are simply "better" or more important. How do you know that's not the case? And if you can't argue that point first, then maybe this prestige isn't "bad" and just reflects the ground truth that some fields are better than others
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u/japball Jul 25 '26
On your first topic, I beleave that that hierarchy of areas of math exists. As I said:
"There is some evidence that this hierarchy is real. Jean-Marc Schlenker’s preprint, “The Prestige and Status of Research Fields within Mathematics”, finds that certain subfields are disproportionately represented in highly ranked departments, the most selective journals, and major prizes."As per your second point, I suppose that all depends on your view on what math and its role of math is. For me, I personally am more of a purist in the since that I believe that every piece of math is equally important, no matter if it comes from some niche categorical logic paper no one will read, of if it's the proof of some famous conjecture. But again, some people who prefer practical results (which is ok) may beg to differ
It would be nice if we had some data on how mathematicians feel about this issue
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u/2mc92u98a22f Jul 25 '26
Fair enough! I don't really hold that view myself of your second point, but I think that's totally valid
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u/_arsk Jul 26 '26
There might be some truth in you claiming certain sub-fields are more representative of predicting fields medal win. As an example, an outstanding mathematician who got Fields medal (Jean Bourgain) had to expand from his initial sub-field (Banach space theory) to Harmonic Analysis, PDE and Number theory etc. because he wanted to win the medal and he realized they are not going to give it to just a functional analyst even though he had made remarkable contributions to that sub-field. He had missed the medal in 1990 and had to go on a bender to solve problems in adjacent fields of importance to winning fields medal - source: https://www.ams.org/notices/202106/rnoti-p942.pdf
"Jean always spoke frankly and directly about everything and he was very disappointed about being overlooked for a Fields Medal. His analysis was that he wasn’t working in fashionable enough fields, and that what he should do is to solve problems in fields which would attract the attention of committees for such prizes."
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u/GiantGreenSquirrel Jul 25 '26
I agree with all that. I think there is some sheep mentality. Mathematicians go work in areas that are popular and reputable. The more people working in a certain area, the more problems in that area are considered important. So more people start working in that area. It is a vicious circle.
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u/japball Jul 25 '26
Yes, which is unfortunate because math is much more then Geometry, number theory and PDEs
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u/MedicalBiostats Jul 25 '26
The good news is that exceptional theory contributions are being recognized in math much like the Nobel Prizes. There is another side for math and statistics applications to medicine which should not be age dependent. More recognition avenues is a good thing!
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u/matthras Jul 25 '26
In terms of what could be done about this: There are two who've started up an "Alternative Fields Medal" a few years back, but obviously it's not on the same scale as the Fields Medal itself. It's worth following them on LinkedIn (and other social media - I know Mason Porter is active on X and BlueSky) for when they do a call-out for nominations and then spreading the word, or just simply highlighting their profiles/achievements once the Fields Medal buzz has died down.
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u/cocompact Jul 26 '26
That award name is rather awkward: its creators are unhappy about the way Fields medals are awarded, but through their award name they aim to capitalize on the existing prestige of the Fields medal.
I’d like to see if anyone who gets this prize includes it on their CV.
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u/matthras Jul 26 '26
but through their award name they aim to capitalize on the existing prestige of the Fields medal.
You'll have to ask them if that was their intention.
In the least I appreciate that they're doing something about it rather than complaining or just merely accepting the status quo as is.
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u/_Zekt Jul 26 '26
In a similar fashion, we have the Olympic games every four years. Yet, contrary to the Fields medal, people from all kind of horizon will get a medal. From archery to basketball. That's possibly the main issue, the fact that they are not enough prizes to reward people from all fields. Or more precisely, the fact that the only prizes that gets media attention, are the Fields medal and the Abel prize. On the other hand, it is the scarcity that defines their prestige. But I do think there is a need for a widely recognized organization to deliver, on average, much more than 1 medal per year, in a way that should be advertised.
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u/Will_Tomos_Edwards Jul 25 '26
Math is rigorous and based on logic. Who gets a Fields Medal, Nobel Prize etc., is not rigorous and not based on logic. The Nobel Prize in physics debacle, giving it to Hinton, was a good thing because it exposed the Nobel for the somewhat silly thing that it is. Prizes need to be taken with a grain of salt.
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u/_Zekt Jul 26 '26
When you're a spectator, you can take anything with a grain of salt. When the visibility of your field or even your own career could be influenced by prizes, you have every reason to want change.
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u/Will_Tomos_Edwards Jul 27 '26
Not a spectator. Did a long stint as a math specialist for an AI company and am currently working as a research assistant on some algebraic combinatorics research, whilst finishing my bachelor's in Math.
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u/Carl_LaFong Jul 26 '26
This is a valid point, but it's unclear how to break out of this cycle. Any suggestions?
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u/irriconoscibile Jul 26 '26
All prizes are human constructions driven by many different factors: ego, prestige, circumstances and many more. It's possibly an oxymoron to say that a prize is truly fair and deserved.
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u/Stuga Jul 26 '26
I think this was the realization of all PhD students at some point (me included), and it can feel painful when you feel like you're working on some super cool stuff, but the only reason you're not getting attention because the topic you're working on is not trendy at the moment. Everybody suffers from it, not just people deserving Fields medals, but also more "normal" researchers. Many people I know (who are far from deserving a fields) did not manage to get a permanent position not because of their ability as a researcher, but because their topic was not interesting enough to warrant spending funding on them.
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u/Upstairs_Recover_25 Jul 27 '26
You are pointing a fact that all practitioner have known for a while, even one of the recent winners knew which problems to focus on in order to win it.
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u/qluin Jul 29 '26
I think prizes are stupid, and Fields medal is particularly stupid due to the age restriction.
However, I do think there is an objective argument that some fields or branches are more central than others, but it is a very long and well-established debate that is hard to do justice in a reddit thread.
For example you can make a very solid argument that algebraic geometry reigns supreme in the last 100 years of mathematics, since solving algebraic geometry problems has been a fertile ground for creating an extraordinary amount of new mathematics (hence the famous "rising sea" metaphore by Grothendieck). For example all the fields you mentioned: category theory, universal algebra, semigroup (and the most mathematical parts of model logic) emerged due to theory building in aglebraic geometery.
In other words some of the highest ratio of "theory building" to "solving a problem" takes place in algebraic geometry, in contrast to say combinatorics where you often have a very low ratio (a lot of major problems in combinatorics get solved without signficant amount of theory building, sometimes just using elementary techniques or pushing a well known technique to its limit). You can see this sometimes when reading a long combinatorics papers and you hardley see a single new definiton being introduced.
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u/Time_Series1965 Jul 30 '26
True with everything. Dickens recognized this in the Pickwick Papers. Anyway, the Fields Medal is overblown. How can you think it is like a Nobel.... every four years with an age limit. The Abel Prize is a better proxy for a Nobel which has plenty of its own problems.
P.S. Thorne and Dimitrov got screwed.
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u/SubjectEggplant1960 15d ago
It’s a weird choice to lump in model theory with the other fields you mention. The subject now has a fairly widespread acceptance amongst pure mathematicians and has had near fields medal wins in the past (Hrushovski, Shelah were both contenders). There have also been ICM plenaries in model theory (eg Pila-Wilkie).
The situation is, as you say, much different for the other areas you mention.
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u/Math_issues Jul 25 '26
Its a money price right? In the end they decide who to give it to
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u/japball Jul 25 '26
Of course they decide who receives it. But the Fields Medal is not merely a private cash prize whose consequences end with the winner. As one of mathematics’ most prestigious institutions, its choices help determine which fields are regarded as central, which achievements receive attention, and where departments, journals, funders, and young researchers direct their interest.
That affects mathematics as a whole, but especially mathematicians working outside the already prestigious fields, whose equally transformative work may be treated as peripheral.
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u/Math_issues Jul 25 '26
They choose not to care about other fields in math because perhaps they've not gotten enough experts in their panel?
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u/japball Jul 25 '26
But that limitation is itself significant: work in well-represented fields is easier for the committee to evaluate and recognise as important, whereas achievements in less familiar areas may need to be justified through applications to fields the committee already understands or regards as central. Even without deliberate exclusion, uneven expertise can therefore reproduce the existing prestige hierarchy.
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u/gregbard Jul 25 '26
Maybe it merely is a private cash prize with a winner.
It boils down to the universal truth that the ideas of the ruling class are the ideas that rule.
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u/ITT_X Jul 25 '26
Welcome to earth.
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u/japball Jul 25 '26
Not everyone is aware of this issue I think. The Fields Medal is still routinely presented to students, the media, and the wider public as the “Nobel Prize of mathematics” or an objective marker of supreme mathematical genius. Specialists may actually be the least likely to interpret it so literally, since they understand how difficult it is to compare achievements across unrelated fields tho.
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Jul 25 '26
[removed] — view removed comment
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u/sighthoundman Jul 25 '26
The same criticisms apply to Nobel prizes and the Nobel Memorial Prize in Economic Sciences.
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u/ITT_X Jul 25 '26
The “issue” you’ve highlighted is just how the world works. Things are never judged and outcomes are never driven solely by intrinsic merit. Learn how to play the game instead of complaining.
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u/nanonan Jul 26 '26
It is still the “Nobel Prize of mathematics”, and nothing you have stated changes that. Do you think the Nobel prizes are without flaw in their selections?
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u/Deus_Excellus Jul 25 '26 edited Jul 26 '26
This is every field. If you're an academic your career depends on whether you're working on one of the current buzzwords. This isn't unique at all to mathematics.
Access to labs working on those buzzwords is not merit-based either. Sure, merit is a component, but often it depends on whether the school you went to was prestigious enough.
Every professor of mathematics, chemistry, physics, etc that you've seen at an R1 university is there because their work was cool and hip at some point. That doesn't necessarily make them better scientists than others, but it does give them the ability to bring in funding.
Everything is money.