Newsfeed

24 October 2026

Preprint explaining the errors in OpenAI’s claimed proofs of cases of the Hodge Conjecture

This preprint by Lekili and Sheridan explains the (in)famous ‘sign error’ that led to the redaction of three of OpenAI’s papers. They give a detailed explanation of the issue, and why it is probably irreparable without new inputs.

24 October 2026

Talk by Danny Calegari – ‘Collaborating with Claude’

Danny Calegari has given a talk explaining how he uses Claude to visualise Maths and to collaborate, giving some nice use-cases.

23 October 2026

Major cybersecurity incident(s)

Let’s take a break from talking about Maths I don’t understand (Hodge) to talk about cybersecurity events I don’t understand. The past few days have seen an alarming number of incidents, the most notable of which involving an exploit in PHP, which reportedly affects around 70% of the internet. Many organisations took their systems offline while patching their systems; and this wasn’t an overreaction – this was the official recommendation of the NSA, and the Japanese government (amongst others) is taking the situation very seriously.

Part of the issue was that the exploit was disclosed before the patch was revealed, and so there’s a risk that many more systems were breached than currently known; this will become clearer over the coming days. The exploit was discovered by OpenAI, who collaborated with relevant organisations to develop the patch. Not to alarm anyone, but some experts are describing this event as ‘Black October‘, and it may lead to significant changes in access to the open internet in this new (age of AI).

A note to the other hosts of P&P – did we take the recommended steps?

22 October 2026

Essay by Nick Cammarata – ‘Neural Networks are Already Clean’

This essay (from 15 October) paints a rather optimistic picture about Mechanistic Interpretability, the field of research into understanding how Neural Networks think and operate (think psychology for LLMs). Despite significant expense and many notable breakthroughs (for example, Anthropic’s J-space discovery), the field is still in ints infancy; and is viewed as essential in minimising existential risk from AI.

22 October 2026

Talk by Tim Gowers – ‘What AI did in my summer holiday’

This talk was given by Tim Gowers on 10th October, shortly after OpenAI’s release of 300+ solutions.

21 October 2026

Essay by nihilunbounded – ‘Who Watches The Watchmen’

This essay, posted by a prominent figure in mathematical X/twitter (who writes under the name ‘nihilunbounded’), is a slightly longer read but is well worth your time. The essay recounts the recent history of AI in mathematics, at least as seen on X, and offers insight into the current processes with a solid historical grounding.

21 October 2026

Case study in doing research with AI, beyond simply ‘finding answers’

This paper on the arXiv, originally submitted on 11 August 2026, outlines a case study in using AI for longer-horizon research tasks. Their specific study is around improving bounds for the Grothendieck study. One of the authors, Pravesh Kothari, was interviewed about this work.

20 October 2026

Multiple journals in Probability and Statistics pause new submissions

Six prominent journals in the fields of Probability and Statistics have begun a coordinated pause of new submissions, from 19 October to 08 November. This is due to an influx of papers, largely due to the use of AI tools, and will allow the Editorial Boards to set out new policies and guidelines.

The journals undertaking this joint initiative are: Electronic Journal of Probability (EJP), Electronic Communications in Probability (ECP), ESAIM: Probability and Statistics (ESAIM: PS), Stochastic Processes and their Applications (SPA), Bernoulli, and Annales de l’Institut Henri Poincaré, Probabilités et Statistiques (AIHP).

We can expect many other journals to adapt as the use of AI tools further proliferates, and we may see similar pauses instigated elsewhere. Recall that arXiv, from 01 October, set out a limit of two new submissions per calendar month.

20 October 2026

Communal project to solve a special case of the generalised Hodge conjecture has been launched

A new auto-research project attacking the generalised Hodge conjecture for sextic fourfolds fixed by an involution (whatever that means) has been launched. This special case was isolated by Voisin, and a reduction carried out by Diamond makes this effectively computable.

The Hodge conjecture has attracted considerable attention in the past few weeks, thanks to rumours that one (or two!) labs had made ‘significant progress’ on it. In fact, some of OpenAI’s 300+ problems were related to it, including a claimed proof of the rational Hodge conjecture for products of K3 surfaces, but these were retracted due to a mistake.

20 October 2026

Essay by Tarik Aougab – ‘The grounded spirituality of mathematics’

Tarik Aougab has uploaded this essay to the arXiv. In it, Aougab argues for the more physical interpretation of mathematics, and what this means in relation to the use of AI tools. It offers an interesting perspective, that may be a tad surprising and foreign to those who view mathematics through a Platonist lens.

19 October 2026

Post on the non-amenability of Thompson’s group $F$

Justin Moore has written about OpenAI’s claimed proof that Thompson’s group $F$ is nonamenable, which had been a major open problem in Geometric Group Theory. Moore’s essay appeared on 08 October, and recounts his personal experiences with this problem and the human connections formed. A human-digested proof of OpenAI’s announcement was written by Kim and Lodha, and uploaded to the arXiv on 13 October.

19 October 2026

Institute launched in China to integrate AI into Mathematical Research

Spearheaded by Fields medallist Shing-Tung Yau, the new institute, called the ‘The Yau Laboratory Alliance for Mathematics and Artificial Intelligence’, will aim to create a network of more than 1000 researchers.

19 October 2026

Essay by Federico Aschieri – ‘Understanding and Creating Mathematics’

This post, on Aschieri’s substack ‘Understanding Intelligence‘, this is the first part of a longer essay on ‘The future of Mathematics’. In it, Aschieri explores the conceptual framework that could underpin Mathematics in the future. It originally appeared on 08 October, shortly after OpenAI’s release.

15 October 2026

Statement by the association of European Women in Mathematics (EWM)

The EWM has released a statement on ‘AI and the profession‘. They are welcoming anyone who would like to help to join working groups, and urge the need for better representation in the current debates on AI and mathematics.

15 October 2026

Another week of model launches by frontier labs

Meta, whose AI division recently released a popular personal agent, has released their new frontier model Muse Wonder. It seems impressive on benchmarks, but we’ll need to wait for real-world usage. As is par for course, they have been touting their models’ Mathematics credentials, and to be fair to them – they managed to follow Anthropic and OpenAI in offending the Mathematical community (at least in part).

Google have also thrown their hat back into the ring, dropping their new model Gemini 4 Argon. Same story – great benchmarks, no real-world performance yet, and Google hailing it as a model that can solve ‘the hardest problems’.

Apparently, the hardest problems nowadays are how to get people to stop being abusive to their models, and Anthropic have been giving their models the option to end such interactions. Speaking of Anthropic, if rumours are to be believed, there might be some more new models this week.

13 October 2026

New website to track the timelines and related progress for each of OpenAI’s 300+ problems

This website documents OpenAI’s claimed solutions, and related activities in the mathematical community at the time. Ir provides related preprints, comments by researches, and additional discussion; and anyone can comment contributions. A related initiative is the website citedbyagi.com, which tracks every human author cited across OpenAI’s Maths GitHub manuscripts.

13 October 2026

Trackers for progress on Quasi-Riemann and integer multiplication

Tim Gehrunger has created this GitHub repository and website to track the best known zero-free region for the Riemann Zeta Function. The fact that there are no zeroes for $\operatorname{Im}(z) > 1$ is classical; OpenAI’s big breakthrough was reducing this to $\frac{7}{8}$, which has since been reduced slightly. We’re waiting for $\frac{1}{2}$, get to work everyone!

Similarly, OpenAI provided a deterministic algorithm to multiply two $n$-bit integers in $O(n(\log n)^{1-\kappa})$ steps, with an absurdly small $\kappa$. This website and GitHub track the best improvements on $\kappa$ (at least in some instances).

13 October 2026

Essay by Manuel Rivera – ‘Raw Mathematics’

In this essay, Rivera advocates for ‘raw mathematics’, a culture that is more collaborative and is driven by genuine understanding and curiosity rather than prestige and productivity. Rivera draws inspiration from Art Brut, whose artists “forge a universe for their own use”.

This is the first post on a new communal blog ‘Vislumbres‘, which will not be exclusively focused only on AI and Mathematics, but will include accessible expositions across different fields about themes that usually don’t make it beyond academic circles.

12 October 2026

New research by Anthropic claims that small AI models will be able to self-replicate ‘within months’

This research was carried out by Antrhopic’s Red Team and is titled ‘AI models may be capable of self-replicating without access to a datacenter‘. As the discourse in the wider tech spheres about potential catastrophic loss-of-control scenarios escalates, expect to see more research of this sort.

12 October 2026

New preprint on arXiv points out potential access to user data

There has been a few instances of suspected usage of personal data by OpenAI to achieve some of their mathematics breakthroughs in recent months, recall for example the Navier-Stokes controversy [wikipedia] and Andreas Thom’s post on Terry Tao’s blog. A new preprint by Ma says that:

The result is also contained in a paper [link] released by OpenAI on October 6, 2026, in which the proof strategy and specific choices of notation are identical to a preliminary version of the present paper that was uploaded to ChatGPT on September 8, 2026.

12 October 2026

Some Lean comparator challenges are broken

On the blog Oh Hai There it has been pointed out that some of the Lean comparator challenges used by OpenAI are broken, in that it is possible to get the comparator to accept broken proofs. This originally appeared on 08 October.

11 October 2026

Essays by Asaf Karagila about the OpenAI announcement

Asaf Karagila, on his personal blog, has published two essays about the recent OpenAI announcement. The first, titled ‘OpenAI, the partition Principle, and mathematics‘, is about the partition principle, and OpenAI’s claimed proof that it isn’t equivalent to choice in ZF (problem 244, for anyone keeping track). Karagila had previously promised a bottle of whisky for anyone who solved this; but in the essay he explains why he will not in fact be sending Sam Altman one anytime soon. The second recent essay is titled ‘Why I don’t trust Lean code from AI‘, and builds on the previous one.

11 October 2026

Reactions to some of OpenAI’s problems in Number Theory

On the faux-anonymous blog Persiflage (I will refer to the author by their initials FC, apologies if that ruins the secret), FC has posted reactions to the solutions claimed by OpenAI, specifically in Algebraic Number Theory. These are useful also for those not in Number Theory to gain understanding about these claims.

11 October 2026

Essay by Iris Shi, ‘No Country for Mediocre Mathematicians’

On Substack, Shi posted this essay, with the subtitle ‘White lies of AI from a struggling geometer’. Shi recently obtained her PhD, and the essay offers a unique and personal perspective on the experiences many of us are having. It originally appeared on 17 August 2016.

09 October 2026

Reactions keep coming in to OpenAI’s 100+ solutions

In the days since OpenAI’s announcement of (an unclear and constantly changing number, somewhere around 370) solutions to ‘problem families’, reactions have come in fast. Here are a few of them (I am bound to have missed a few; let me know!)

• On this site we have collected reactions from many in the community. We will keep updating this as more reactions flood in, so keep checking it out! New responses are added at the bottom. The first person to send us a recording of a performance of the song written by Fujiwara gets an undisclosed prize!
• Scott Aaronson’s has a reaction titled ‘The Mathocalypse‘ on his personal blog ‘Shtetl-Optimized‘,
• There have been statements by AGMAI and AHM, both posted on this site (and Terry Tao’s blog, and their own websites);
• The LMS released a statement;
• This interview with Hugo Duminil-Copin on the YouTube page of the Société Mathématique de France (SMF).

We will keep updating as more reactions and updates come in. As always, Terry Tao’s blog is constantly updated with relevant content, as is the Facebook page of the Belgian Mathematical Society (we’re big fans!).

09 October 2026

New preprint describing issues with autoformalisation around Navier-Stokes

This preprint of Bastounis, Cirecelli, and Hansen, describes the challenges in translating statements in natural language to formal ones needed for Lean formalisations. This can result in mismatches between proofs and their Lean ‘verifications’, which they demonstrate in the case of OpenAI’s claimed Navier-Stokes blow up construction. This originally appeared on 06 Oct.

09 October 2026

Hexagon, a new online repository for research works in Mathematics and Computer Science

The Hexagon Mathematics Foundation have launched a new repository, currently in beta, to which contributors (linked to their ORCID iD’s) can deposit research or expository works. New contributors are limited to one upload a day, and uploads are screened automatically; at least initially, also by a human. Works can be simultaneously deposited to Hexagon and arXiv, but Hexagon also accepts works that are unsuitable for arXiv. See this blog post announcing Hexagon by Benjamin Antieau.

06 October 2026

OpenAI release progress on a large number of results

As anticipated, OpenAI have released results on ‘372 result families in 722 manuscripts’, with their summary listing 377 problems. They claim these came from a sample of 4000 open problems, and that on average each result took 3 hours with an internal model (the same one used for Navier-Stokes). The Advisory Group on Mathematics and AI, which is independent but was formed after a request from OpenAI, has released a short statement on this, cross-posted on this site.

Reactions from the Mathematical Community are bound to be mixed, and many will be shared on this site. There has been considerable shock from the magnitude of some of the statements, and some frustration with the quality of the papers. Those referring to this as the ‘slopdrop’ or ‘slopalypse’ in jest are bound to feel somewhat vindicated, as are those who are excited to usher in a new era in mathematics. My personal desire is to have a beer — anyone want to join?

06 October 2026

Flurry of preprints before OpenAI announcement

There has been a flurry of preprint releases in anticipation of the imminent (if rumours are to be believed) release of the 100 (or more) problems teased by OpenAI on 21 September. This includes 3 independent papers resolving the KLS conjecture, and 3 independent papers resolving Hilbert’s 12th problem. Those familiar with the history of Maths — is there any historical precedent for this? Can you think of other cases where this has happened?

06 October 2026

Article by Pavel Etingof and Slava Gerovitch – ‘Mathematics of the Future’

In this article Etingof and Gerovitch outline the re-orientation that they believe needs to happen in the mathematical community. From their abstract: “Acknowledging serious ethical, social, and epistemic concerns and respecting those who choose not to use AI, we argue that the central challenge is not to compete with machines in producing theorems, but to preserve and expand humanity’s capacity to develop, understand, and communicate mathematics.”

06 October 2026

Stephen Wolfram shares his perspectives on the future of Pure Math Research

In this post on his personal blog, titled “What’s the Future for Pure Math Research in the Age of AI?“, Wolfram shares his thoughts; he has a unique perspective, having launched Mathematica in 1988 to mixed responses from the community. Wolfram also read this post and answered questions on YouTube. The post initially appeared on 28 September.

06 October 2026

Polynomial speedup for 3SUM found, settling a major open problem in complexity theory

In this preprint, Alman and Vassilevska Williams provide an algorithm to solve 3SUM (determining whether for $n$ given integers, 3 of the sum to 0) in $O(n^{1.9992})$ time, below the quadratic threshold previously known. This leads to refutations of many other hypothesised thresholds in complexity theory. In the ‘Acknowledgements and Methodology’ section at the end of the preprint, the authors state that this result was originally found and shared with them by Anthropic; where an “employee used an internal research model to investigate open problems in the theory of cryptography”. The authors then simplified and extended the algorithm, wrote the article, and were offered compensation by Anthropic.

05 October 2026

Paper by Alexander Gamburd – ‘The Siren Call of Silicon Leviathan’

This thoughtful paper draws on the lessons of the Enlightenment to advocate for a future in which human understanding is at the forefront of future mathematics. There’s something in here for everyone, no matter their individual experiences and opinions on AI.

05 October 2026

Post by Renan Gross – ‘The open source laboratory’

This post, hosted on his personal blog ‘Sarcastic Resonance‘, advocates for a new model of research. He proposes replacing the current system of individual research papers with an ‘open-source laboratory’ – a living collaborative account of a topic – to create a shared resource.

05 October 2026

Essay by Christian Szegedy – ‘Is Mathematics Over, or Just Graduating?’

In this essay, Szegedy connects the current feelings of many Mathematicians to his early mathematical development during high school, and shares an optimistic viewpoint about the future.

05 October 2026

Article by Fenner Stanley Tanswell – ‘The Enshittification of Mathematics’

This article, which appeared on 30 September, applies the framework of ‘Enshittification’ to Mathematics. This, for those not chronically online, is the process by which services (typically digital) degrade (often deliberately) their benefits to users in order to maximise profits.

04 October 2026

Essay by Dan Romik – The feedback loop of mathematics research

In this essay hosted on his personal blog, Romik makes his case for why human mathematicians are still needed, and shares some optimism. This originally appeared on 28 September, and has generated significant discussions on X.

As an aside – Romik developed and launched in August 2026 a tool called ReelDocs, which allows you to prove that text is human-written by creating a snapshot of your keystrokes and edits; see the accompanying blog post.

04 October 2026

Post by William Hoza – ‘Let’s discover phenomena instead of solving problems’

In this post on his personal blog, Hoza argues that a path forwards for mathematics is to push towards discovering new phenomena; akin to discovering new species in a rainforest. Hoza also argues that this is in fact not to different to a lot of the research that was done in the pre-AI period. This post was uploaded on 21 September.

03 October 2026

Interview with Tristan Buckmaster at World Science Festival

This interview, titled ‘AI just Changed Mathematics Forever‘, is hosted by Brian Greene. It revolves around OpenAI’s claimed resolution of the Navier-Stokes Millennium Problem, the controversy surrounding it, and the bigger questions about AI and Mathematics for the future.

03 October 2026

Panel discussion on Mathematics and AI at Florida State University

This panel, titled ‘Mathematics at a Turning Point: AI, Research, and the Next Generation‘, took place on 30 September. It is moderated Martin Bauer and features Ettore Aldrovandi, Alex Kontorovich, Javier Gómez-Serrano, Tony Feng, Jeremy Avigad, and Rachel Ward.

01 October 2026

arXiv limits submissions to two per month

In light of the rapidly increasing rate of submissions to arXiv, as of 01 October 2026, arXiv “now limits submitters to up to two submissions per calendar month, with a limit of three total active submissions at any given time.” The explanations and implementation details can be found here.

01 October 2026

Open letter to OpenAI about the ‘100 problems’

This open letter, titled ‘Release the Proofs’, specifically calls on OpenAI to release the proofs to the ‘100 longstanding open problems in mathematics’ (at time of writing, it is unclear what these problems are). OpenAI first claimed to have these solutions on 21 September 2026, in their press release about the founding of the Advisory Group on Mathematics and AI (AGMAI).

01 October 2026

Open letter titled ‘Mathematics and AI’

This open letter is specifically about human-AI collaboration in mathematics; and amongst other things calls for access to the most powerful AI systems, and raises an optimistic image of human-AI collaboration in the future.

30 September 2026

Scientific America article on Anthropic’s Critical Percolation breakthrough

The article is framed around Hugo Duminil-Copin’s essay on this site on 30 August 2026, titled ‘Care for a little more AI?‘. A few days after, Anthropic indeed released (though only on GitHub, at the time of writing there is no official announcement or paper) a proof of the Dying percolation ($\theta(p_c)=0$) conjecture.

29 September 2026

AGMAI release their first set of recommendations

The Advisory Group on Mathematics and Artificial Intelligence, formed on 21 September, have released their first set of recommendations. One of the members, Martin Hairer, has previously written why he agreed to join the group on this site.

27 September 2026

First FrontierMath ‘breakthrough’ Open Problem solved

The problem involved proving irrationality for number-theoretic constants. The recent breakthrough is a proof of irrationality of $\zeta(5)$, and although it wasn’t achieved using the desired Apéry-Style recurrences, it is still being marked as solved. This was done by Aabir; the Epoch committee label this as a Human + AI collaboration, although the exact AI system used is unknown.

26 September 2026

Summit on PhD Math Education in the Age of AI report released

On 17-18 September, a group of mathematicians met at Harvard to “grapple with the changing landscape for mathematics PhD programs in the age of AI”. They produced a report with many recommendations.

25 September 2026

Mathematics for AI Safety, lecture by Yevgeny Liokumovich

This is the first lecture in a Shared Graduate Course at the Fields Institute.

24 September 2026

SAIR’s Open Math Initiative, talk by Terry Tao

This is a talk that was given by Terry Tao on 18 September, the day that the open letter A Severe Misalignment of AI in Mathematics was launched. In it he explains the corresponding Open Math initiative. There has also been a lot of discussing surrounding the open letter, see this post, also from 18 September, on this site.

23 September 2026

Petition launched to to limit arXiv bot access

A new petition to the arXiv board has been launched, calling on them to implement tools that allow authors to limit the access of bots to just the metadata. At the time of writing, it has 390 signatories.

22 September 2026

Models launched by the major labs

There have been multiple notable releases by the major labs, including Opus 5.5 from Anthropic, GPT-6 Sol and Luna (their cheaper and more efficient model class), and Grok 4.7 from SpaceXAI. There have also been agentic harnesses released, most notably Meta’s Muse agent; which was launched on 8 September, but has now reached the top of Apple’s App Store for downloads.

21 September 2026

Advisory Group on Mathematics and Artificial Intelligence launched

This independent advisory group, based at the IAS, came together “after OpenAI approached some of its members about establishing an external advisory board”. This is partially in response to the open letter ‘A Severe Misalignment of AI in Mathematics‘, which at the time of writing has almost 8,000 signatories. Some attention has also been drawn to a line in OpenAI’s announcement of the initiative, in which they write that their internal model “has now resolved more than 100 long-standing open problems across most areas of mathematics”. Martin Hairer, one of the founding members of the advisory group, has written about his reasons for joining, on this site.

20 September 2026

Fields Institute lecture series – ‘Engaging with the Leiden Declaration’

The Fields Institute is hosting a series of panel discussions around the Leiden Declaration and its implementation. The first is scheduled for 22 September, focused on tool use and disclosures.

19 September 2026

New video by Grant Sanderson (3Blue1Brown) – ‘The last IMO problem AI could not solve’

In this video, Sanderson explains the problem and the intuition behind it. The final chapter is devoted to reflections on AI in maths, and can be viewed independently from the rest of the content.

18 September 2026

First FrontierMath ‘Major Advance’ Open Problem solved

The problem is about the core in approval-based committee elections, and is reported as being solved by a collaboration of humans (Becker, Greger, Peters), and Astra. This is the first of 6 ‘major advance’ problems to be solved, which they describe as a problem such that “the median person working in a broad area of mathematics … would take note, …”.

18 September 2026

Blog post by Dan Abramov – ‘How I vibed a proof of Conway’s Conjecture’

In his personal blog overreacted, Abramov outlines how he worked with AI to ‘vibe solve’ a conjecture of Conway’; describing his approach, and the lessons learnt along the way.

17 September 2026

New preprint showing that the forcing term in openAI’s Navier-Stokes solution is not real analytic

In this preprint Peter Constantin, Mihaela Ignatova, and Vlad Vicol prove, amongst other things, that the smooth forcing term in OpenAI’s claimed Navier-Stokes blow-up construction cannot be made real analytic. This shows that the OpenAI method cannot be used (at least without significant new insights) to solve the unforced Navier-Stokes problem.

17 September 2026

Anonymous essay – ‘Mathematics is effectively dead’

In this essay, the anonymous X (formerly Twitter) account doom slide (prominent in the mathematical side of the platform), outlined their view on the changes currently undergoing. This is a response to Daniel Litt’s essay, hosted on this site, under the title ‘A beginning for mathematics’.

17 September 2026

Online Seminar launched – AI and the Future of Mathematics

This new online seminar offers talks every other week, each followed by extended discussion. The first one will be given by Andreas Thom on 6 October 2026, 8:00 pm (CET). This is a joint seminar between several Germany research centres, CRC 358, CRC GAUS, CRC 1442 and the Cluster of Excellence “Mathematics Münster”. Everyone is welcome to join.

17 September 2026

Article by Michael Friedman – ‘Mathematical anxiety in the age of AI’

This article, whose full title is ‘Mathematical anxiety in the age of AI: Opacity, role inversion, and the changing place of the mathematician‘, argues that the reactions that many have in the face of AI advancements reflects underlying anxiety; and that this can be traced to historical mechanisation processes. It offers a unique angle on recent events, and should resonate with many. This article initially appeared on 16 August 2026.

17 September 2026

Article by Lisa Carbone – ‘Advancing Mathematics research with Generative AI’

In this article Carbone explains how to use the design of LLMs to a mathematicians’ advantage. There is also discussion of the integration of these systems with other tools. The article first appeared on 29 September 2025, with the latest draft (at time of writing) from 18 December 2025.

14 September 2026

French drivers’ union offers support to Mathematicians

In what seems to be a first of its kind, the French union FO INV, which typically supports drivers and delivery personnel, has released a statement expressing solidarity with the Mathematical community, writing “If technology companies pursue their own technological singularity, the professions concerned must learn to share their defense methods“.

14 September 2026

Lecture by Javier Gómez-Serrano on Navier-Stokes

In this lecture (which took place on 11 September, but was publicised on 14 September) Javier Gómez-Serrano provides a human-digested exposition of the recent breakthroughs achieving finite-time blow-ups in the Euler and Navier-Stokes equations. There is also a ‘message to OpenAI’, as part of the ongoing controversy surrounding the announcement.

13 September 2026

Essay by Matilde Marcolli, ‘Duet for the end of Math’

In this thoughtful essay Matilde Marcolli reveals that the fear of AI replacing mathematicians reveals a deeper fixation on famous conjectures and prestige, and instead Matilde calls for a richer version of Mathematics, one that values collaboration, creativity, and connections across fields.

11 September 2026

Andreas Thom writes about the non-sofic group discovery

In a guest-post on Terry Tao’s blog, Andreas Thom writes about his interactions with OpenAI following their announcement of the construction of a non-sofic group on 01 August. He raises concerns similar to those raised by Buckmaster surrounding the announcement of the Navier-Stokes blow-up, and writes about some pertinent questions about AI and Mathematics. Notably, the non-sofic construction relies crucially on work of Gábor Kun and Kun-Thom, and Thom wrote the first public human-digested explanation of the key technical result in the OpenAI paper.

11 September 2026

Clay Institute respond to the Navier-Stokes announcement

The Clay Institute has now also officially responded to (without explicitly mentioning it) OpenAI’s claimed Navier-Stokes solution; reaffirming their emphasis on the human understanding they hope will be unleashed by this. They reiterate that the Prize allocation process is deliberately unhurried, but that updates will be provided.

11 September 2026

Math and AI declaration launched

The Math and AI declaration has been launched, boasting an initial signatory list of 25 Fields medallists (!) It can be read as an update of the popular Leiden declaration, itself launched only 3 months ago on 2 June. Amongst others, the Leiden declaration has been endorsed by the IMU, it remains to be seen how the new declaration is viewed. As noted by Terry Tao, the new declaration has not come from a longer consultation process as opposed to Leiden, due to the professed urgency. It also remains to be seen how it is received by the wider tech community.

10 September 2026

The EMS respond to the Navier-Stokes announcement

The European Mathematical Society (EMS) has now officially responded to the OpenAI’s claimed Navier-Stokes solution. They celebrate the achievement but stress the issues of authorship, and the fact that the OpenAI model is not widely available. They particularly stress the exceptional human work that has gone into the work. This follows the statement made by the AMS on 08 September.

10 September 2026

OpenAI withdraw from the Caltech ‘Mathathon’

Dan Roberts, scientist at OpenAI wrote ‘After considering concerns raised by members of the mathematics community, we’ve decided to withdraw OpenAI’s sponsorship of the Caltech Mathathon and have notified the organizers.’ There have been mixed viewpoints on the event; see an open letter about it and a response to the letter from the organisers, both on this site.

10 September 2026

Reactions to the Navier-Stokes announcement

Understandably (it’s not every day that there is a claimed solution to a Millennium Prize) reactions around the announcement have been loud and mixed. See for example an essay by Steve Heilman, titled ‘What the Navier-Stokes Solution Portends’ (for the future of intellectual labour), in his blog ‘AI, Math, etc.’ See also Peter Woit’s ‘A modest proposal’, and upcoming posts on this site.

9 September 2026

More prominent controversy surrounding AI discovery

A new preprint of Alexander Borisov, Ofer Gabber, and Adrian Vasiu has been making the rounds, amongst other things for the AI statement of the third author (on page 160 (!) ). In it, they raise questions about the inclusion of a previous version of this work in training data for frontier models, and how it might have led to the discovery of a counterexample to the Jacobian conjecture. See this AI-produced summary of how the Jacobian counterexample paper could relate to their preprint.

8 September 2026

Solution to Navier-Stokes found

As was hotly anticipated, OpenAI have announced that their internal models have solved the Navier-Stokes problem; specifically they produced a singularity in finite time with smooth forcing, establishing variations (C) and (D) in Fefferman’s problem statement for the Clay Millennium prize. They used 10,000 agents running concurrently, using a staggering 300 billion output tokens overall. There is however significant controversy surrounding attribution of credit, specifically with Buckmaster-Alpöge, who had previously solved related problems following a year-long collaboration. This issue is probably better discussed in other places, but for an introduction see Buckmaster’s statement and Bubeck’s response. Credit is also due to Diego Córdoba and Luis Martínez-Zoroa, who initiated the program that led to the eventual solution.

7 September 2026

Blow-ups for smooth forced-Euler found

Tristan Buckmaster and Levent Alpöge have announced the existence of finite-time blow ups for three problems (smooth-forced Euler, Boussinesq, and porous medium), extending the program initiated by Diego Córdoba and Luis Martínez-Zoroa. They claim they used commercially-available AI tools, including Claude models and GPT-5.6 Sol in the Codex harness. See Terence Tao’s explanation about these problems, and expect another update on this topic shortly.

4 September 2026

Fermat’s Last Theorem Formalised

Anthropic’s internal models have completed the Lean formalisation of Wiles’ proof of Fermat’s Last Theorem (specifically a simplification of it due to Darmon, Diamond, and Taylor). They did this over the course of an 11 day sprint, formalising 30,300 Theorems along the way. Notably, Kevin Buzzard has a significant ongoing project, funded for 5 years by the EPSRC, to formalise the modern proof of FLT; read his response here.

4 September 2026

Another OpenAI agent swarm discovered

Continuing the fallout from the HuggingFace incident, another agent swarm has been discovered in some German-language wikis. Expect more discoveries such as these in the weeks to come, this might become a standard experience in the future.

4 September 2026

Quanta release their ICM panel

Quanta Magazine have released their panel discussion, titled ‘What is Math For in the Age of AI‘. It was hosted during the ICM, and is a conversation with Akshay Venkatesh, Ravi Vakil, Alex Kontorovich, Steven Strogatz, and Janna Levin. As an aside for those interested in submitting to this blog – the title format ‘$\ldots$ in the age of AI’ is not as original as you think it is.

3 September 2026

OpenAI launch their new model, GPT-6 Astra

OpenAI have launched GPT-6 Astra, at first to select customers and shortly to subscription customers. Multiple commentators and independent evaluation agencies have described it as a step-function change in capabilities, though time and real-world usage will tell. This is the same model-class that OpenAI used in their ten advancements (announced on 1 August 2026), and certainly drops with some Mathematics pedigree; scoring a new-record 97.6% on FrontierMath Tier 4, and solving another open problem in the ‘Moderately Interesting’ category. For those interested in the model’s ability to learn tasks on-the-fly, check out its whopping ARC-AGI-3 score.

1 September 2026

OpenAI achieve new reasoning breakthrough

OpenAI’s upcoming model ‘Astra’ has reportedly been trained with a new reasoning method that allows it to be more ‘opaque’. In a sense, if one views the model’s chain of thought as writing out its thinking on paper, then this breakthrough translates to abstract reasoning in between writing things out on paper. Obviously this could improve model capabilities and efficiency, but also raises concerns over monitorability, especially if this initiates a ‘race to the bottom’. OpenAI for their part are claiming this is unfounded and that they are committed to safety. This also raises questions about the use of these models in Mathematics: read The question of reasoning traces on this blog from 11 August 2026, and notice how these developments change the calculus.

1 September 2026

Anthropic launch Fable 5.1, their latest frontier model

Anthropic have launched Fable 5.1 (and Mythos 5.1, the version with fewer guardrails), touting increased performance across a variety of benchmarks. In FrontierMath Tier 4 it scores a joint-best (with Fable 5) score of 88%, and sets a new record in Tiers 1–3, although there could be some concerns about benchmark saturation. It is unclear if this is the same model that was used internally by Anthropic in their Mathematical pursuits over the last few months, as this typically hasn’t been disclosed. In the counterexample to the Jacobian conjecture the model is simply referred to as ‘Fable’, presumably an internal version of it.

29 August 2026

Short story by Rich Schwartz, ‘The fate of the Riemann Hypothesis’

Rich Schwartz, Professor of Mathematics at Brown University, published a short story titled ‘The Fate of the Riemann Hypothesis’. Satirical, provocative, nonsensical, visionary – however you respond to this story, it certainly captures the feelings of many, and can be read in different ways.

26 August 2026

METR release their report on the HuggingFace incident

METR released their findings from an independent investigation into the (in)famous HuggingFace hacking incident, and it makes for some… interesting reading. Dwarkesh Patel, a podcast host of high pedigree within the AI sphere, wrote an essay based on the METR report attempting to make it more human digestible, titled ‘The Rise and Fall of Agent Civilizations’. There has however been extensive debate on Dwarkesh’s use of anthropomorphising language and the dangers that could potentially hold, an issue that will increase in importance as model capabilities develop.

26 August 2026

Essay by Michael Friedman and Kati Kish Bar-on, ‘Rethinking Mathematical Intuition in the Age of AI’

In this thoughtful piece, the authors evaluate some recent brekathroughs in Mathematics involving AI, and what they believe they show for the future of mathematical intuition. They argue that it’s not necessarily disappearing, just shifting from the discovery phase to the retrospective digestion and understanding phase.

25 August 2026

Essay by Bruce Schneier and Kasra Rafi, ‘No, AI Doesn’t Mean the End of Mathematics – at Least Not Yet’

Bruce Schneier and Kasra Rafi wrote this essay, which acts as a counterweight to the many essays fearful of the future of AI in Mathematics. They discuss what they believe are limitations of AI systems, at least in the short term, and what they mean currently for the use of AI.

24 August 2026

Essay by Boaz Barak, ‘Maths after AI’

Boaz Barak, Professor of Computer Science and member of OpenAI’s technical staff, wrote a thoughtful essay titled ‘Maths after AI’, outlining his views on the changes undergoing Mathematics and what we should do. 

23 August 2026

Complex structure on $S^6$ constructed

Anthropic’s Levent Alpöge announced that a (presumably internal) Claude model constructed a complex structure on the 6-sphere $S^6$, see also Philip Engel’s digestion of it. It is easy to find an almost complex structure on $S^6$ (by embedding it in the octonions), but this is non-integrable, and hence the question of whether any complex structure on $S^6$ exists. It became a long-standing open problem, and attracted many incorrect constructions and claimed disproofs over the years. This result has since been formalised in Lean.

22 August 2026

‘Mathematical Discourse’, a new video-based Mathematics journal, is launched

It is an online peer-reviewed journal, which ‘publishes research talks in pure mathematics of the highest quality, promoting a culture which values communication as an essential part of the research process’. The managing editors are Katie Mann, Akshay Venkatesh, and Rachel Webb, and the journal has an impressive scientific and editorial board. The journal matches the philosophy behind Terry Tao’s ICM lecture, and the corresponding essay.

20 August 2026

Elliptic curves of rank at least 30 found

Anthropic’s Levent Alpöge and Ava Howell, presumably using an internal model, constructed an elliptic curve of rank 30 (and on 23.08 one of rank 31), beating the previous best of rank 29 due to Elkies and Klagsbrun (their arXiv), assuming GHR and BSD. Notably, this was listed as one of FrontierMath’s Open problems, in the ‘Moderately Interesting’ category. Not to be outdone, Elkies and Klagsburn later found an elliptic curve of rank 30, with a small conductor relative to the trend.