100+ reactions to 100+ solutions

Tim Santens

The results of OpenAI are almost unbelievable, Quasi-Riemann is a result that I thought was complete science fiction. Much can be said about the behavior of the AI labs, but given the existence of these models mathematics as a research discipline will have to undergo a massive transformation.

Roman Sauer

A common piece of wisdom is that mathematicians only have a few tricks in their bag – which they skillfully apply again and again. I have such trick whose unreasonable effectiveness surprised me a few times: replacing a manifold by a certain measurable foliated object, a construction that comes from ergodic theory, and using it in unexpected contexts.

More recently, I relied on this trick in a paper with Sabine Braun on macroscopic scalar curvature, which was further extended by Hannah Alpert. This trick and these two papers now appear in OpenAI’s deeply creative solutions of problems 207 and 335 – problems so different, they could not be farther apart!

I am stunned.

What do I take away from this night? We cannot defensively narrow our focus to what we currently regard as human qualities in mathematics. This will force us into a diminishing role. Instead, we have to radically expand our view of what a mathematician is and does.

Xiaolei Wu

It is the last day of a week-long national holiday here in China. We had just taken a short family trip to a nearby coastal city. After waking up, I followed my usual routine and checked my phone. I was surprised to find many messages containing the same PDF file. Apparently, OpenAI had solved many open problems.

So I took a look at the group theory section. It looked unreal. These were almost all the major open problems in my field! How is it possible? So I sent a message asking where the file had come from. The answer was that OpenAI had just officially announced it.

I checked the group theory section again. Yes, these really were almost all of them. So Thompson’s group F is non-amenable. OK, I had tried it many times using ChatGPT, but I guess I didn’t have the newest model or enough tokens. Eilenberg–Ganea is false—not so surprising, perhaps, but how? OK, at least the Whitehead Conjecture has not fallen. And what? They had completely solved the Boone–Higman conjecture? And the ambient simple group could even be of type ($F_\infty$)$\ldots$ I had expected this day to arrive, but maybe something like one year later.

After a while, I started to look at other sessions, starting from topology. So the Borel conjecture fails in dimension 4, the coarse Baum-Connes also fails$\ldots$

The rest of the day felt long. More messages came in. My social media feeds were flooded with discussions about the list. Some people were still claiming that the problems solved in their fields were actually not so important. But I guess I had already passed that stage.

Nicholas Williams

We’ve certainly learnt a lot of new facts from this release. At the same time, it will take a while for us mathematicians to digest all of these papers. There’s no doubt that large language models have changed things enormously, but the end goal must always be human understanding. It feels important to point out that mathematicians do more than just solve open problems from the literature: we also have to grapple with the subject matter and think about what the good conjectures should be in the first place. But on the other hand, it does feel like the end of an era where the ability to solve problems was a distinguishing feature of being a mathematician. And this feels sad for those of us who like to solve problems.

Henry Bradford

Today my chief feelings are disorientation and bewilderment. It’s as if the pace of events has outstripped us by such a margin, that it is impossible to know how one should respond, or even how to start finding out how to respond. My greatest worry is for the prospects for bringing through the next generation of mathematicians: we know how much time and effort it takes to refine one’s craft in our subject to the point where one can meaningfully contribute to the advancement of knowledge, and I’m anxious that the necessary incentives may no longer exist for bright young people to invest that effort. I am holding on to hope that a bright future for the practice of mathematics is possible, in which our understanding of the mathematical universe is enriched, rather than undermined, by the ubiquity of AI tools. To get there though, we need as a community to agree what a realistic pathway should look like for a young scholar to establish themselves in our subject.

Henry Wilton

There are many problems on OpenAI’s list that I care a lot about: the Cannon conjecture, the non-residually-finite hyperbolic group, the Boone—Higman conjecture and at least ten others. I haven’t looked at any of them in detail yet, although a cursory glance suggests that the write-ups vary wildly in quality. It will take months of work to go through the proofs of all of them, to check if the proofs are correct and to try to digest them.

But I think it’s important to focus on the big picture. If OpenAI wanted to destroy the mathematical community, this would be a great way to go about it. Open problems are a resource that the mathematical community developed over decades or even centuries. The value they have is the value we have given them. They provide structure, a yardstick for progress and long-term goals. The advent of AI was always going to be a seismic shock, but this huge dump of papers is a tsunami that we have no time to prepare for. It’s clear that OpenAI will happily wash away our community’s structures to further their financial interests.

There is a long list of mathematical structures and norms that OpenAI seems to be intent on damaging. By refusing to name the authors of their papers, they imply that mathematics is no longer a human endeavour. They don’t submit their results to journals, which both implies that the norms of the mathematical community are defunct and makes it impossible for the community to digest their results. OpenAI haven’t shared fundamental information about how the problems are selected or the failure rates of their attempts. This information should be a necessary requirement for any serious scientific discussion of their tools. And of course there is the basic financial injustice that their hugely valuable models learned to reason by copying the very reasoning that our community made available for free through the Open Access movement.

I hope the mathematical community can adjust our practices and norms quickly enough to survive the shock of these developments, but I am fearful.

Macarena Arenas

Perhaps the AI companies could, if they wanted to, do something to benefit humankind and the planet we live on (ending world hunger, solving global warming, curing cancer…), but this is not it, and it is not being done for altruistic purposes.

Regarding the mathematics, some of it is unverified, most of it is badly written, and (probably) none of it is transparent in documenting how it was produced, who was involved, and what was its cost (in every sense). All of it is of course impressive. Some of it might come to benefit mathematics and mathematicians, but overall I think it is an example of a corporation trying to exert manipulative pressure on a field of knowledge, and I think it has been done irresponsibly. I hope that we as a community can find our way through this mess, but for the time being, it is a mess, and it causes more harm than good. It will distract a lot of people from working on their own ideas; it will disrupt many existing research agendas; it will create conflict in the mathematical community; it will encourage a lot of opportunistic, ill-informed use of these technologies (thus adding to the “slop” and creating more noise for us to navigate through); and it will discourage many, many people from doing mathematics.

Johannes Schmitt

One thing to keep in mind about this drop of papers is that they were not created by OpenAI with the main goal of making progress in mathematical research, or even primarily for PR (there were some PR posts, but e.g. the final announcement was comparatively restrained). The main reason these papers exist is that OpenAI used unsolved math problems to evaluate their internal models for the purpose of improving their capabilities. They had to use hard open problems because these are the only ones still posing a challenge to frontier AI models. I believe that, from OpenAI’s point of view, the papers are essentially a byproduct of their internal development.

Thus the company could just have remained silent, or picked a few flashy results for another blog post. I think it is to their credit that instead they reached out to the math community for guidance on how to proceed, leading to the formation of the Advisory Group on Mathematics and Artificial Intelligence. They followed some of the advice by this group, such as a timely release and (partial) Lean certificates, but not all of it, and in particular announced that they will continue to use open math problems for model development. In my view, the most pressing need of the math community right now is to have a central platform to discuss the results, start digesting the (by multiple accounts from colleagues, terribly written) papers, and report any mathematical mistakes or missing attribution. An OpenAI-controlled GitHub repository, with issues and pull requests disabled, cannot serve this purpose, and the ball is in OpenAI’s court to move the papers to a neutral repository and to back such a community effort with the promised funding for workshops and special programs.

Lvzhou Chen

To be clear, I only read some parts of the overview and a few papers in the collection without understanding the full details. I know most of the problems in group theory and quite a few in topology. I spent a few years thinking about the Kervaire conjecture, Howie’s conjecture (Problem 256) and related problems.

I have some rather mixed feelings towards this. On the positive side, I would like to learn and digest the new techniques or insights that lead to the solutions, and I hope to use them to better understand or solve other problems. On the other hand, I feel that solving so many problems in such a short time may damage the math community and profession. Often time, new discoveries were made in attempts to solve open problems (like the ones solved here). So we could have lost tons of opportunities to make other new discoveries; I hope what is worth discovering will be discovered sooner or later. In any case, this certainly creates even more stress and uncertainty than before for math people, especially junior ones (e.g. PhD students): How would the profession change? What should PhD students focus on? How should we train them? I wish what I called “damage” can be a part of reforming the profession in an eventually positive way, and (young) people interested in math do not get discouraged because of this.

As for the future, I guess (or hope?) that there will be a limit of the effect of AI: the landscape of math stabilizes after a dramatic change, and we obtain a new sense of difficulty. After all, we always find new interesting and harder problems after seeing the solutions to old ones. However, it might take quite some time before it stabilizes (if it does), and before then, it could be quite chaotic. Probably one can focus more on some brand-new areas and problems that seem fundamental and important. Even after it stabilizes, it is still a question whether human contributions are needed to make new discoveries. If it becomes unnecessary, one can still focus on better understanding the new discoveries, but I personally might find it less interesting and would rather work on other stuff. Maybe the question is not whether humans can still contribute; instead, as Ruixiang puts it somewhere, will mathematicians still have the courage to work on the remaining ones that are difficult in a new sense?

I guess most of these will just turn out to be silly and naive thoughts, but my understanding of this post is to record honest feelings, silly or not.

Enrico Fatighenti

I am not a Luddite — quite the opposite. I have used, and still use on a daily basis, AI as a bibliographic tool, to test random ideas, to speed up bureaucracy, as a helping hand with teaching, and so on. I do not use it to generate proofs, but I am not judgemental about people who do.

However, I was genuinely pissed off by today’s announcement.

To be fair, my first reaction was almost boredom. Yes, the AI has “proved” (really? Are we sure? Can we even understand what is written there?) a bunch of interesting results in my field. Not even a Millennium Problem. Pff.

But the more I thought about it, the more annoyed I became. What bothers me most is the double standard.
Whenever I referee a paper, or receive a thesis or project from a student, I apply the same standards that most of us do. If a paper or thesis is so badly written that I cannot get past the first page without considerable effort, I reject it and ask for a new, readable version.
With these AI-generated papers, I feel that we, as a community, are not applying the same standards of quality and rigour. They can simply release multiple 100+ page papers claiming to have solved this or that problem, often with redundant arguments, unclear logical structure, multiple dead ends, and strange or unsettling terminology — in other words, slop. And we are then expected to go through it, check it, clean it up, simplify it, and explain what is actually going on.

This comes at a considerable cost to us in terms of time and effort, while they can simply move on and slop-bulldoze the next conjecture. And, of course, the credit remains theirs. In some sense, we are willingly contributing to our own demise.

I think the paradigm has to shift. We should apply to AI-generated mathematical announcements the same standards of rigour, clarity, and exposition that we demand from human-generated papers. Do you want the approval of the mathematical community? A badge of legitimacy? Then give us something that is actually readable and verifiable. If not, we should simply ignore it.

In other words, I do not think that we, as a community, should spend our time cleaning up their mess in order to increase someone else’s commercial revenues.

It is going to be difficult to win this game, but at the very least we can try to change the rules a little, so that the game is not completely skewed in their favour.

Giulio Tiozzo

My feelings today are a mixture of excitement and worry: it’s nice to see many problems solved, and to discover new proofs. Yet it’s clear that our profession has changed forever. Some thoughts:

1) I don’t think the main problem of AI-generated math proofs is that they are not understandable: all first solutions to big problems were hard to process. The problem is that no one would want to do it, as they fear they would not get any reward, either by the community or on a personal level.

2) What puzzled me a lot is the method of communication: After AGMAI was created, the way I imagined the current math drop worked was: Each of the 9 experts chooses a problem and explains it carefully in a video, or writes a nice companion paper. Instead, it’s just a massive GitHub drop of slop papers. What did we even need AGMAI for?

3) It’s clear many papers were never reviewed by a human: for instance, I took their example #254 (an Artin group with no CAT(0) action): their group has 116 generators, but after feeding it to Astra, in 10 minutes I got a new example with 12 generators. Clearly, they were in a rush to post the big drop, prioritizing quantity over quality…Why?

This does not solve the misalignment between labs and academics at all; indeed, the question remains: are frontier labs our friends, or our enemies?

Raphael Appenzeller

This timeline is crazy. I feel conflicted. There are many good reasons for and against the use of AI in mathematics, and the uncertainty about the future only grows.

Alexandre Martin

After yesterday’s announcement, many of us are left stunned and unsure of how to adapt to this new situation. Already, there are calls from colleagues to organise international reading groups and workshops to make sense of it all and to write a “human account” of some of these claimed proofs. In my case, while some of the results have hit pretty close to home, and while some of these colleagues are very much well-intentioned, I have decided not to take part in any such initiative.

First of all, I do not wish to be part of what in some cases would amount to free publicity and free refereeing for OpenAI, contributing to the false narrative being pushed that this company is now actively collaborating with the mathematical community. A second reason is that in organising ourselves in such a way and at such a scale, we implicitly accept this new division of labour, where proof-production can become completely decoupled from any form of proof-comprehension, and where this dehumanized proof-production process becomes increasingly tied with an arms race for computing resources. I think this is unsustainable on many levels, and it is also simply not the future that I wish for our community and for our shared mathematical practice.

Baptiste Serraille

After seeing the huge advances posted today, I decided that I needed to try out ChatGPT extensively for the first time. Most of the time I do not give it my own ideas, I am too afraid of them transiting through ChatGPT to other users or OpenAI itself and losing my grip on them. Thus, I decided to try it on open problems of the fields or on projects that I will not have time to try on the short/medium term. Eventually, one of the problem did fall and it seems that a project that I had envisioned might also give something. I am both amazed at the results and the technological advances and scared at the fact that the pace is much higher than I can keep up with. I am also not happy to write a nice paper about ideas that are not mine and do it mostly because I believe that the field will benefit from it.

Matt Zaremsky

For several years now, the most major component of my general research program has been the Boone-Higman conjecture in group theory, including a series of papers joint with many others in which we’ve been gradually setting up and fine-tuning a nice sufficient condition for a group to satisfy the conjecture. I’d say roughly once per year for the last 4 years we’ve made some sort of big breakthrough that’s pushed the program forward in a strong way. Now this AI company decided the Boone-Higman conjecture had gotten famous enough that they should spike the ball, so they did. It turns out our sufficient condition always works, and the main mechanism we were missing was morally quite similar to the thing that was our 2025 breakthrough (extremely roughly speaking, the key is affine-like actions of things). So, apparently we were really on to something, and it’s believable we could have gotten there in another year or two.

So, how does that feel? Well, it feels like we’ve been working on an archeological dig for 4 years, gradually discovering more and more of a really cool-looking dinosaur skeleton, and then a trillion-dollar company showed up and just blasted the whole thing with TNT, handed us the whole skeleton, and walked away. So, I guess I’d say, “not great”.

Ilya Kazachkov

Firstly, beyond the scorched earth that the announcement leaves behind, its magnitude indicates that the changes to our profession are likely to be very profound. There are many questions covering all aspects of our work that we, as a community, will need to answer. Many of them have been raised in the media, including on this blog: What does it mean to do a PhD in mathematics? How will publishing, evaluation, hiring, and funding work? What is the role of mathematical research in society?

As for research, the most intriguing question the AI revolution seems to bring is in what constitutes a “good” mathematical problem. In particular, we will need to understand what types of problems, if any, can be solved by a human (and AI), but not by AI with little human intervention.

In turbulent times, panicking is the worst strategy. Whatever answers we inevitably come up with, after a period of uncertainty and anxiety, things will settle. I believe we will adapt, and the profession will live on in the new normal.

Mark Hagen

I have received emails from early-career colleagues who have asked my advice on other matters in the past, who currently seem very understandably distressed by these developments and the uncertainty they create, and are asking for advice again (or for some reassurance, or something). Today, I have prioritised trying to figure out what to say to them over reading any of the OpenAI preprints in detail, so I don’t have any useful comments on the content. I still don’t know how to answer those requests for advice, either (yet).

But irrespective of how interesting we might find some of these developments mathematically, I think the manner of their release reveals we’re clearly confronted with a bad-faith (human-institutional) actor here, whose interests are not aligned with my understanding of the point of scientific research (or any other worthwhile human pursuits).

I am worried about people that do mathematics, about my friends and colleagues, and about cultural practices (like doing mathematics) to which I attach importance. I’m even more worried about AI possibilities that are more general than mathematics: economic dislocation, degraded collective human capacities, supercharged exploitation and ecological collapse, new and terrifying means of repression and violence, etc. Mathematics has a reputation in our culture for being difficult and intimidating; evidently OpenAI have decided this renders our shared human endeavour a suitable vehicle to hijack as a display of power.

In any case, it seems like a situation in which it’s very easy to get confused or overwhelmed. This is a community that takes pride in being relatively tightly-knit and able to have productive collective discussions. This seems to be a test of those hypotheses.

Cyril Houdayer

Like many of my friends in operator algebras, I was shocked by the scale of the results announced by OpenAI. Two of these problems are particularly dear to my heart, because I have spent so much time thinking about them and working on them.

Connes’ rigidity conjecture was one of my favourite problems. Part of the sway it had on me, until the late 2010s, was due to the fact that it seemed to be completely out of reach. Even Sorin Popa’s deformation/rigidity theory did not offer a way to tackle it. In joint work with Rémi Boutonnet, we began studying higher-rank lattices using von Neumann algebraic methods. Our noncommutative version of the Nevo–Margulis–Zimmer theorems provided a conceptual framework for trying to recover the rank of a lattice from its von Neumann algebra. I pursued this direction through noncommutative boundary theory, uncovering some interesting rigidity phenomena and pushing hard towards rank recovery.

I did not solve the conjecture, but I had come to believe that a frontier model from OpenAI might do so, and I had been mentally preparing for that possibility. I had made my peace with it. Now I am genuinely thrilled by the announced solution, and I want to understand the proof. I look forward to gathering my PhD students and postdocs to study it together during the IHP programme « Operator Algebras: Approximation, Rigidity and Dynamics ».

Working on this conjecture meant a great deal to me, even though I did not solve it. It led me into the mathematics of Furstenberg, Margulis and Zimmer. I made new friends and discovered connections between operator algebras and discrete subgroups of Lie groups. Those experiences remain meaningful to me, and I am excited about the new connections that may emerge.

Connes’ bicentralizer problem has a different history for us, and there is an important clarification to make. This problem has played a central role in the structure theory of type III von Neumann algebras. Amine Marrakchi and I began working on it more than ten years ago, and Amine subsequently developed a substantial collection of tools and techniques to attack it. Recently, we returned to the problem from a rather indirect direction. In our joint work from September, we solved it for all type III_1 factors. The solution emerged as a consequence of a result about the bicentralizer of a flow on a type II_1 factor, within a broader classification theorem.

The proof announced by OpenAI builds on the methods developed in Amine’s earlier work and on the resonance mechanism we discovered together. Our recent work settled the nonrelative version of the bicentralizer problem. OpenAI’s result extends ours by establishing the relative version.

These announcements have shaken our community. I understand why many colleagues, especially younger ones, find this moment difficult, and my excitement about the mathematics does not make that unease disappear. I hope we can talk openly about both feelings. For me, the immediate response is to learn the proofs, discuss them with students and colleagues, and keep exploring the mathematics together.

Danny Calegari

About 15 years ago my experience with the lack of interest of the mathematical community in what I was discovering in the theory of stable commutator length in free groups (see https://www.quantamagazine.org/how-failure-has-made-mathematics-stronger-20240522/) led me to a resolution: from then on I would only work on what interested me. If I wanted to prove a theorem I’d prove a theorem. If I wanted to draw a picture I’d draw a picture. I’ve paid a (small in my opinion) price for this in the sense that I don’t think many people read my papers. On the plus side this means that OpenAI didn’t solve any problem that I was working on. They settled a couple of questions that I was curious about, but not enormously, and not in a way that was tied to my own sense of self.

The project that I am currently most excited about is the theory of higher dimensional zippers, and the most exciting aspect of it is that it’s now become possible to draw pictures (really: animations). One thing I love about some parts of mathematics (the Langlands program is an example) is that it’s not just a collection of theorems and conjectures; it’s a framework, and a story. I’ve always tried to look for the underlying story in my research (at least for the last 15 years). OpenAI hasn’t ruined that, at least for me (yet). Another example: a framework like Sullivan’s dictionary is more important than the top 100 papers in holomorphic dynamics (including Sullivan’s papers). In my own research, following the lead of Sullivan’s dictionary, I have just discovered the analog in the holomorphic dynamics world of a finite depth foliation: it’s a grafting of carpet wheels! Even naming this is exciting for me! Writing down the definitions and proving the theorems is the next step, but the reason (for me) to do that step is so that I can write some programs and draw pictures/animations.

Anyway: people do math in lots of different ways for lots of different reasons. I honestly think there will be more ways to do math in the future, not fewer. For the curious, the following movie (https://math.uchicago.edu/~dannyc/gallery/cs_zippers_movie.mp4) is what I am currently ecstatic about; it’s a zipper in S^3 associated to an arithmetic complex hyperbolic lattice, following some ideas of Isenrich-Py. The zipper is linked, and this reflects the fact that there is a circle valued Morse function with critical points but they are all in the middle dimension (2). I love this! I am currently collaborating with a musician to set it to music. It was produced by code that was written by Claude, generalizing code (and an algorithm) that I originally came up with in the 2d setting. If that’s not worth celebrating I don’t know what is.

Vadim Alekseev

I got really impressed by the resolution of the free factor problem (the von Neumann algebras of free groups are all isomorphic to each other), since I had been wrong about the expected outcome: I thought they rather should be non-isomorphic, since the parallel evidence from measured group theory (in my opinion) rather pointed there (via L2-Betti numbers); also, simultaneously the fact that the von Neumann algebras of SL(n,Z) are non-isomorphic justifies this line of thinking! So to me it is exciting: we now have to figure out what exactly still remains parallel between measured group theory and operator algebras, and what diverges and why.

Tristan Humbert

I woke up this this morning to an email of a collaborator informing me that Open AI announced a proof of Katok’s entropy conjecture, a three-decades old open problem which was the subject of my PhD and more generally the main motivation of my research. The conjecture was settled by Katok for surfaces and mostly open in higher dimension. The main project of my thesis was to prove a local version of the conjecture near complex hyperbolic metrics. After this I was planning on attacking the conjecture more globally. Open AI claimed a proof of the conjecture in full generality this morning and my first impression was sadness to see my favorite problem get “killed” by AI ; it is definitely easier to ignore a problem when it does not affect you directly. Next, I felt stressed because I am currently applying for postdocs and more of my research plan was now obsolete and I spent the whole morning rewriting everything and sending emails to my advisors in panic. Finally, I felt anger after opening the paper and observing that it was mostly unreadable slop. The sane reaction would have just been to ignore the paper but I think I care too much about the problem to act as if it did not exist. The only conclusion I can make right now is that Open AI’s standards for mathematical publication is far too low and is harming mathematical research maybe in an irreversible way and that the mathematical community, motivated by genuine scientific curiosity, is accepting to work for free in order to “validate” these unreadable proofs.

Tim Gehrunger

I was very curious to finally see the statement drop from OpenAI. I am not sure what they could have possibly done to meet my expectations, but I was at first slightly underwhelmed by what was in there, especially given the results in my own field of arithmetic geometry.

Of course a lot of the other work is impressive, and in some fields of mathematics such as combinatorics several of the main conjectures of the field appear to have been resolved, likely changing the way these fields will go in the future.

One thing that I found striking is that many of the articles were not as polished as one may have liked, with articles citing removed supporting manuscripts, missing citations of relevant prior work alongside minor mistakes (that I would strongly expect an AI to catch). This is something that a more elaborate workflow with dedicated agents for review, correct attribution and literature acknowledgements would have very likely caught. I hope future releases will use such a system to improve polish and to make sure that attribution for earlier results is given.

Peter Scholze

We should remember that we are all in this together; mathematics is a marathon, not a sprint; and the goal is and always will be the human understanding of mathematics, which will invariably take time.

What is worrying me the most, at the moment, is actually the (in)security of cryptography. Finding algorithms breaking standard cryptographic protocols is a number theory problem whose difficulty does, to my non-expert eyes, not significantly exceed what these systems are now capable of. And it would have disastrous consequences on society if such an algorithm is found.

Tasmin Chu

I was working on the $p_c < p_u$ problem. I first learned about it in undergrad. There’s a beautiful 1996 paper by Benjamini and Schramm where they conjectured that Bernoulli(p) percolation on every nonamenable quasi-transitive graph has a phase with infinitely many infinite clusters. The converse is true by work of Burton and Keane. I loved this problem so deeply. It said something so beautiful and simple about these highly treelike graphs which had somehow eluded proof in the general case. It’s why I fell in love with percolation theory. Unsurprisingly the OpenAI preprint builds on work of my advisor Tom Hutchcroft and the approach he laid out to me a year ago.

My grant proposal due next week has to be rewritten. Maybe grant proposals don’t even make sense anymore. I have related results and work that I can talk about instead which were part of an overall research program I had to attack this problem. But I’m afraid to even talk about ongoing work now or do a research “announcement”. I have a high enough profile now I believe it’s not unlikely someone would adversarially try to prompt my result into completion.

I realize now that what I wanted more than to know that $p_c < p_u$ is true was the time and space to think about it for the next few years. I feel a profound sense of grief for the working conditions I believed I would have. No one can take away the beauty and value of mathematical thought from me, but they can effectively disempower me in my own profession. What hurts the most is that OpenAI doesn’t even care about this result. They don’t know that our community, our intellectual thought, our knowledge is a living thing. To be frank, I find myself absolutely disgusted by the society I live in.

Terence Tao

My feelings on recent developments are very mixed and complex.

On the one hand, many of the AI-generated proofs appear to introduce clever new ideas that will be fruitful once digested, while also building upon the existing contributions of countless human mathematicians past and present. But at the same time, I am deeply frustrated that, in sharp contrast to traditional breakthroughs, none of the humans involved in these proofs are available to take questions, give talks, attend conferences, submit papers to journals, train students, or otherwise participate in the subsequent development of these results.

Similarly, I am excited by the possibility of the community being able to use these tools to tackle ambitious and large-scale projects that one could not have even dreamed of in the past. But I am horrified by the many person-years of ongoing patient and deliberately slow research efforts – particularly by graduate students and postdocs – towards many motivating problems in mathematics being casually disrupted or destroyed by such a release. Much as one cannot unhear a movie spoiler or a crossword clue, one cannot explore a problem as profitably and richly once one is aware of an existing solution. Yes, one can still analyze and digest such an answer; but the best opportunity to do so is at the moment of its discovery, and such moments are increasingly wasted when delegated entirely to AI tools.

And I mourn the path not taken, and the opportunities lost in the frantic race to develop this technology. Labs submitting their frontier models to independent researchers for proper scientific evaluation. Coordination with the research community to ensure these tools are applied to complement and enhance the abilities and activities of human researchers, rather than compete with them. Use of these tools to foster collaboration and sharing, rather than competition and secrecy. Opening new doors, without closing old ones.

But that is not the path we now find ourselves in. Instead, the community needs to come together more than ever. To clearly declare our own standards and values, to build our own tools and practices, to support our most vulnerable members, and to chart our own path forward. Let’s get to work.

Elia Fioravanti

While a feel a degree of excitement at seeing resolved some problems I considered almost inapproachable, this is overshadowed by grief and resentment at seeing a tombstone placed over many of the most promising and exciting directions in our field, where human solutions were well within reach in just a couple of years. I hope those working on those problems don’t give up on their work, we could still learn a great deal from it.

I also hope our community come together to digest the new results, senior and young mathematicians alike, regardless of attitudes on LLM use. The goal should not, however, be the writing of preprints containing insights developed through this process, unless the endeavour is supported with substantial funds contributed by AI companies.

Srivatsav Kunnawalkam Elayavalli

I was initially surprised when I saw the resolution to the free group factor problem. I thought that the free group factors ought to be non-isomorphic, and tried hard for a number of years to prove it in this direction. In any case, I am now fully convinced that theorems/proofs have almost entirely lost their currency in the profession. We will have to prioritize human understanding, and device methods to reward and incentivize this. Repeatedly posting AI generated and verified manuscripts achieves nothing but frustration and disorientation. As I said in my previous Proofs and Prompts article, my goal in this profession is to enhance my manodharma. Right now I am having plenty of enjoyment, I am spending all my time preparing the course I am teaching on “free C*-algebras”. I have excellent students who are very interested in the material. I have no interest in derailing myself and engaging in AI-induced indigestion by starting to read the tsunami of results from OpenAI at this moment.

Yang Li

Assuming that these proofs are correct, I must say that I am very impressed about these results, and if these results are properly understood, they may greatly accelerate the progress of maths. My main concern is sociological in nature. In particular, I feel that PhD students and postdocs are the most vulnerable group in an age of radical change, and the community needs to find a way to protect the younger generation. I also think that given the power of the technology, computational resources should be made more equally accessible.

Michael Chapman

I was amazed by the scale and breadth of the recent results coming out of OpenAI in their announcement. A few problems, such as the refutations to the Kaplansky conjectures, the existence of non-residually finite hyperbolic groups, bounded degree coboundary expanders in all dimensions, and the unique games conjecture were all problems “I grew up on” – namely, I learned about them quite early in my studies and was fascinated by them for years (and also worked on some of them extensively). I mostly want to sit down and read as much as I can, to sort out the main ideas and to study the results. This is quite humbling, and as a young researcher also disorienting. Nevertheless, I am more excited than afraid, and hope the mathematical community at large, and my own research community at the smaller scale, will grow stronger out of this.

Constantin Kogler

11 days ago, I witnessed GPT-6 Astra solve one of the best-known questions in my area. Excited by the stunning proof, I teamed up with a longstanding collaborator to rewrite and rethink the solution. After numerous days of hard work, we had a nearly finished paper by Monday. The same conjecture was claimed as Paper 148 by OpenAI on Tuesday. My collaborator wanted to publish on Tuesday, yet I wanted to check some aspect of the literature more thoroughly.

After the initial shock of the announcement, I personally didn’t feel strongly affected by the claimed solution of OpenAI soon afterwards. The key idea was anyhow by Astra. We published our paper on the arXiv on Wednesday. Had the main idea of the paper been our own, it would have been by an order of magnitude our best result. I was excited that we were involved in mathematics of that level and were probably the first humans to understand the solution. I hope others will write their own viewpoint on the proof, so that we will have several expositions with different perspectives.

Paper 153 is also on a question I deeply care about. I am sure I can write a clearer explanation than OpenAI did. So I look forward to understanding what happened and to expressing, hopefully with collaborators, our viewpoint on this profound piece of mathematics.

Yes, machines with such capabilities will change everything. Not only for mathematics but also for the world. I have sympathy for many of my colleagues being upset by this, especially those who already had stunning results themselves or outstanding work in preparation. For myself, I can’t help but feel like living in a mathematical wonderland with the next field-defining idea being discovered whenever I am ready for it.

Andreas Thom

Right now I feel proud that the mathematical community laid the foundation of such a terrific development. Let’s see how it feels tomorrow; we have to answer some serious questions. But in any case, we have enough to read for the coming winter.

Anna Chavez Caliz

Today I’m standing in a more optimistic side. We had a very stimulating and engaging conference last week, here in Cuernavaca. It was clear to me, more than ever, that an essential part of our job is not to be alone in an office, in front of a blackboard, or writing papers for only a small fraction of the population. I appreciated having the chance to remind myself that we still have the power to be excited about math when we go out and talk to others. As Tolstoy said, the enjoyment lies in the search for truth, not in the finding it.

Tobias Osborne

I have been closely following LLM capabilities for a good year now, and thought I was more or less acclimated (numb) to the pace of progress. Although I am not completely surprised, it is hard not to feel more than a little overwhelmed by OpenAI’s mathdump today. It is impossible to properly unpack all of this in a paragraph, especially so early, but I wanted to record a couple of thoughts: (1) So far I have only looked at a couple of the contributions close to my heart. I am struck by how recognisable the individual parts are. The wilder creativity lies in their counterintuitive composition. I would probably have given up on these combinations, or talked myself out of trying them. I am extremely curious to hear what experts make of the more significant problems. One feature I looked for seems absent: undecipherable “alien artifact” ideas and methods. The arguments are presented in prose and, although rough, seem approachable. (2) I think it is the right call to simply share all this stuff in the open: cognition is now abundant, and we should lobby for this resource to be made freely available as widely as possible. (3) I would caution that LLMs have in no way “solved all of mathematics”, any more than they have “solved all of software engineering”. Their capabilities are very spiky, and if you use them regularly, you know what that means. Undeniably, things are going to change: typing code into a computer with your fingers already feels anachronistic, yet engineering remains challenging. There is so much more to the job of a mathematician than solving problems. I am optimistic that the era of “Big M” Mathematics and “Big P” Physics has now well and truly arrived: we can finally take much more ambitious steps and reconsider the big questions that have driven our fields for centuries. Maybe now we can work together to make meaningful progress on them in our lifetimes.

Mahan Mj

Was going through the non residually finite group construction put out by OAI today, and it looked to me that there are really two quite different modules in the proof. One purely algebraic and the other geometric. After fiddling around with Astra for some time today, it gave me a reasonable sounding purely group-ring theoretic criterion for non-residual finiteness of a group. It suggested that this might also certify that the group is non-sofic. I have gone through the AI generated proof briefly, but not yet had time to check it thoroughly. At any rate, it looks like this is the one piece of the construction that is relatively new in the sense that it builds on the AI generated non-sofic group from some days back.

What does seem to be a pity is the following. By and large, the community has not yet had time to really absorb the non-sofic construction from about a month ago. It is quite conceivable that by playing around with that example a number of such criteria would come up over time from different hands. This could lead to a charting of the largely uncharted territory of non-sofic/non-residually finite groups. A consequence would be clarity and understanding–two of the main human reasons for doing mathematics in the first place. The present breakneck speed of things without human understanding compromises precisely this.

Barna Saha

I’m worried how big corporates are controlling academic research. In one case, an Anthropic employee asked a famous academic from a top university to sign an NDA, offered compensation and authorship to verify a result that many researchers have spent decades working. This is unthinkable in academic research. Authorship to work does not come in this way. In other incident, OpenAI dumps solutions to 772 problems in Math and adjacent fields-many of which are outstanding breakthroughs. General academics don’t have access to their powerful internal models. Publicly available models do not even come close. This creates a huge gap in accessibility and equality. I am seeing a lot of frustrations among students.

Josh Frisch

For years now, whenever I’ve met somebody new at a conference, I’ve asked them: “If you could solve any one question, what would it be?” The goal was to get beyond a list of “famous” problems and find out what mathematicians truly cared about. What did we really want to know? Only one person has ever answered “the Riemann hypothesis.”

Like many other mathematicians, my main emotion thus far in 2026 has been loss: loss of meaning, loss of purpose, loss of the era of human proofs, loss of the ability to picture the future. With the release yesterday, October 6, 2026, of hundreds of beautiful results—many, maybe most, answering someone’s “one question”—I am trying to move beyond loss. There are so many beautiful results here: problems nobody had any approaches for, algorithms nobody thought could possibly exist, unexpected isomorphisms, constructions and proofs. A mathematics built from the echoes and scaffolding of human mathematics, but one that clearly will go beyond it.

There are, there must be, so many beautiful ideas in this deluge of proofs. If we can learn the answer to our one question, even if the proof did not come from us, even if it did not come from any human being, then we need to try to understand it and to share that understanding with each other.

David Fisher

No human being should have to respond this quickly. Maybe AI can inspire a slow math movement. If I were cleverer, I would write this as a haiku.¹

Martin Bridson

The scope of last night’s announcement is truly breathtaking. Until very recently, I would not have imagined that the frontiers of mathematics could move so far in one day. Beyond the remarkable list of problems that appear to have been solved, I am deeply impressed by the diverse forms of reasoning sketched in the documents that accompany the announcement.

These announcements will be exciting for many, frightening for others, and devastating for some.

Let me start with the case for excitement. Where there is excitement, it will surely derive not from the closing of open problems but from the opening of new possibilities. In the short term, there will be a flood of enhanced human understanding as the global community of mathematicians absorbs, refines and enhances the arguments produced by the machines. In some cases, experts will kick themselves for having missed a connection between disparate parts of the literature; in other cases, they may be amazed that an apparent trick works and will subsequently advance their fields by unearthing a new phenomenon that explains it.

In all cases, the struggle to wrestle human understanding from the machines’ formalities will unleash greater ambition for what we can achieve (working with AI agents) in this new era of mathematics. Some of our favourite mountains have been conquered, but behind them are bigger mountains that we can tackle with new equipment.

At the same time, we must resist the temptation to believe that digging insights out of announcements from AI labs will become the paramount task of our time. The global community of mathematicians has to be steadfast in its resolve to decide for themselves which research directions merit the most attention. We should embrace the power that the machines offer, but we should not be indentured to follow their lead.

This image of servitude brings us to the fear that undoubtedly stalks alongside the excitement associated to the ascent of AI’s ability to do mathematics.

The list of problems covered by OpenAI’s announcement includes several that were guiding challenges in my own research. I regret the loss of these guiding problems and I am convinced that future announcements will rob me of many others. I am saddened by this loss but not devastated. This relatively sanguine reaction undoubtedly reflects my career stage; I would have been less sanguine twenty years ago. I am acutely aware that today’s announcement will affect the lives of younger colleagues more profoundly.

Personally, I am looking to learning the new ideas and constructions hidden in the AI-sketches of the new results, and I am particularly looking forward to engaging with colleagues from all career stages in group efforts to understand what has been done. I anticipate a community effort to add layers of human insight and explanation. I think this effort will be genuinely communal and I think that we are going to have great seminars!

Nevertheless, I also have to confess to a nagging worry that a way of life that I have loved, in which the joy of the hunt for new discoveries was front and centre, will morph into something less familiar and less viscerally appealing to me. Profound understanding has always been the driving motivation of the research mathematician. In the hard struggle to glean understanding we are sustained by the joy of discovery and there is great personal joy to be had from understanding something beautiful for oneself. But for me, and I suspect for most of us, there is a greater joy in discovering and then sharing something that is new to humanity. If the role of a typical mathematician were reduced entirely to explaining the output of others (humans or machines), this greater joy would be lost and being a mathematician would be a less appealing vocation. This may not be our future, but it is a legitimate fear.

What is beyond doubt, I think, is that the recent developments have profound implications for the structure of our profession. We have to adapt our structures quickly, particularly with respect to the apprenticeship stage of our profession — PhD and postdoc years. Beyond that, there are many issues of credit and recognition, the role of publishing etc.

I also share the general unease about the lack of alignment between the commercial interests of AI labs and the values of our global community. There is an inherent and fundamental tension that cannot be resolved without sustained, robust engagement. We certainly cannot entrust the future well-being of mathematics to their goodwill.

Konrad Wrobel

Scrolling through the list of claimed results, I alternated between shock and apathy repeatedly. The sheer quantity I can only say is exciting (even outside of the standouts I personally care about and the others I’m familiar with), even if it is simultaneously incredibly emotionally draining. It feels surprisingly anticlimactic capstoned by the mountains of work we have in front of us in parsing these, frankly horrifically written, papers and what the new ideas are inside. I can only guess how long it will be before I can internalize the ideas relevant to me.

Alon Dogon

My feelings on the matter have changed so many times throughout the day, ranging from severe fear for the future to excitement for having finally answers for many great problems.

If I had to pick one personally, the equivalence of strong Ulam stability and amenability (along with Diximier’s problem) has particularly touched me, as I have spent several years thinking about it seriously.

The solution seems to combine Fursternburg’s boundary theory with quantum circuits from quantum computing, how wild is that?

In general, it is impressive to have many conjectures settled in the positive this round.

Hugo Duminil-Copin

I expected that one day we would be surpassed, and that it would happen systematically. But yesterday’s announcement hit with a force I had not anticipated. Dozens of papers deal with topics I was working on. Between results that beat you to the finish line and thousand-page proofs, I don’t even know where to look anymore.

Not a single one of the major open problems I have publicly mentioned throughout my career (whether in a talk, a lecture, an article, or even a grant proposal) was left untouched by the announcement. Everything has been claimed to be proved.

I expected to see a few of them in the list. But not all of them. Not all at once. Not with such nonchalance.

“For the glory of the human mind,” they said…

The shock is immense. I am paralysed. Tomorrow, we will find a way forward. We will rethink our profession and how we work. We are a resilient community, and I have no doubt that we will adapt. But for now, I simply don’t have the energy. I think back on all those years, all those faces… I think of my colleagues, my students… And I fear I won’t be able to find the right words.

Petra Schwer

We have talked about “the list” throughout the day with many people at the workshop I am at. I am feeling lots of mixed feelings today. Ranging from shock to a certain degree of amazement about everything the technology can do. I am also angry that we are being bombarded with ‘solutions’ by companies that seem to have little to no interest in the actual content. They seem happy about the dramatic headlines helping them to gain better funding. They don’t seem to care about everything being shaken up so fast that we (the community of mathematicians) can no longer keep up with cleaning up the mess.

Something that worries me is the small changes I am already seeing in my own behavior. I am no longer as open as I was in the past when talking about my research projects and plans. I did, for example, not answer freely to some of the questions after my talk. This is not just me. People are becoming more cautious. Mistrust is spreading, and that is not good. I am lucky to be part of mathematical communities that largely trust(ed?) each other. Seeing that change worries me.

What worries me even more is seeing the junior mathematicians around me struggle. Today I also saw a lot of fear. How can I help them stay afloat?

Is mathematics dead? Clearly, no². What is happening to us right now is definitely a massive shake-up, earthquake, storm. There are a lot of questions to be addressed. For sure the mathematical research landscape will change. How exactly? I have absolutely no idea. And I very much hope that the communities (and people) I care for will come out on the other side with only a black eye.

Bryna Kra

There are deep and far-ranging results in this release, giving us a view on the powerful tools that now exist for exploring mathematics. But math is about more than producing theorems and this method of release loses so much along the way. Understanding this work is an enormous undertaking, and unlike work produced just a few months ago, there is no one to ask when we get stuck in a proof. This is not part of the culture of mathematics.

As a community, we have to come together and work to keep what we value. Our goal of understanding mathematics has not changed, but the methods of getting there have. One of my concerns is the ecosystem that allowed the body of work being used now to make the advances is being destroyed. The mathematics community has mostly been collaborative: we share questions, talk about work in progress, and give others ideas on how to approach a problem. By making it easy to translate those parts of our work into proofs, we short-circuit the understanding that is needed to have impact. This way of releasing results closes off directions of research, rather than opening new vistas.

The math community is already coming together to hold deep discussions on how to navigate this time of turbulence and change. It is time for us to move from discussion into implementation, charting the course for the future of our profession. The training for a doctorate, the hiring of junior faculty, the evaluation for promotion, the criteria for publication, the modes of publication all need to be scrutinized and updated. The good that comes out of this situation is the fall of the nonproductive traditions of our community, while keeping the parts we value.

Alvaro Lozano-Robledo

The “Big OpenAI drop” is nothing short of historic, possibly the single most important day in the history of mathematics thus far. Many of the problems with now proposed solutions in the Oct. 6, 2026 drop would represent huge contributions to their respective fields: quasi-RH, the second part of Hilbert’s 16th, Hilbert’s 10th over Q, the Hodge Conjecture of CM abelian varieties, Goldfeld’s conjecture, fast integer multiplication… They are undeniably huge contributions.

And yet, they have not changed my mindset. On the contrary, we already knew their models can do amazing things (e.g., Navier-Stokes). We already knew the frontier models can connect dots in the existing literature in ingenious ways (e.g., unit-distance conjecture). We already knew that OpenAI can spend a mind-boggling amount of resources to attack problems. We also know the price for their top-level subscription is about to increase significantly, up to $500/month.

Also, we suspected that their models have limitations, and the new release shows evidence of that too. In their report, they mention that they attacked 4000 open problems, and their model was able to make progress on about 700 related problems. Yes, some of the ones they were able to solve are huge. But it also shows that their models are limited in some ways: their goal was RH, not quasi-RH. Their goal was the full Hodge, not Hodge for CM varieties. Their goal was BSD, not Goldfeld’s 50-50. Again, quasi-RH is huge! But it is not RH.

Are any of the solutions using new ideas that are outside of the convex hull of the current ideas in the literature (in the sense of Nestor Guillen)? We will need mathematicians and time to digest these new proofs and understand what connections are being made, and whether brand new ideas were actually discovered in the process. There is a lot of mathematical research that remains to be done with and without the aid of LLMs. What has changed is that now there are new mountains of mathematics to explain and communicate to others.

Julian Wykowski

Navigating today, I kept thinking about a passage in Stanisław Lem’s Solaris (1961), where the main character fantasises about the existence of a bóg ułomny. This has been translated into English as an imperfect god, although I believe a more faithful translation would be a defective or disabled god. The passage reads:

“I’m not thinking of a god whose imperfection arises out of the candour of his human creators, but one whose imperfection represents his essential characteristic: a god limited in his omniscience and power, fallible, incapable of foreseeing the consequences of his acts, and creating things that lead to horror. He is a … sick god, whose ambitions exceed his powers and who does not realise it at first. A god who has created clocks, but not the time they measure. He has created systems or mechanisms that served specific ends but have now overstepped and betrayed them. And he has created eternity, which was to have measured his power, and which measures his unending defeat.”

Many members of the community agree that the main purpose of open questions in mathematics is to guide theory building and produce understanding, rather than a binary answer to some problem with limited applications in the real world. In that sense, we truly have created systems or mechanisms that served specific ends but have now overstepped and betrayed them. While it is certainly in OpenAI’s marketing interests to spread a narrative that mathematics has been “solved” through the existence of some lean code on some server, I sincerely hope our community will not succumb to such a defeatist narrative. Instead, I hope that we will find a consensus-based, organised approach to adapt our work to this new reality, in ways that align with our values, support our pursuit of human understanding, and benefit the construction of mathematical theory. This may well include embracing AI, but only in a form that maximises its positive and minimises its negative impact on the aspects of mathematics we consider fundamental. In the meantime, if OpenAI wants everyone to believe they are a deity, it is our duty to remember how defective their idea of deity is.

Ben Green

I was not expecting the magnitude of some of these results. Most particularly, seeing a proof of no zeros of Dirichlet L-functions to the right of $\mathrm{Re} s = 7/8$ (and a second, short, proof of no Siegel zeros) is absolutely shocking to me, but there are many other breathtaking advances. Closer to my particular expertise, many of the central problems of additive combinatorics have fallen, including around three quarters of the aims I had for an ERC Advanced Grant, awarded only in June. The work of understanding these solutions and the associated context properly is significant and, from what I can glean from the current manuscripts, likely to be very worthwhile. I’ll start with number 182, which shows that any subset of $\{1,\ldots,N\}$ of size $N^{1 – c}$ has two elements differing by a square.

Sam Hughes

It was a privilege of a lifetime to get to do research level maths. But not like this. How much beauty have we lost?

Emily Riehl

Firstly, kudos to whoever is behind the website citedbyagi.com, which recognizes the mathematicians whose work is cited by the manuscript collection released by OpenAI. It will take quite a while to understand what exactly has been achieved there. But whatever it is was only possible because mathematicians formulated the conjectures, proved the surrounding results, and shared their ideas – in conversations, talks, expository writing, and papers – so that other humans, and now AI, could learn from them. I hope they continue, because like many others I love learning new mathematics from other humans and always will.