AI4Science field
Research releases and public discussion in this scientific field, ordered by time.
[This is a guest post by Sophie Huiberts , being an expanded version of a post on her own blog . This blog post was initially written in a different file format and converted using AI. — T.] When you prove a theorem, you are imparted a responsibility to the field of mathematics. At minimum you need to explain what you did, including which steps required new ideas and which steps are already well-known. More substanti
By Melissa Lee , Monash University In August, artificial intelligence company OpenAI announced ten advances in mathematics and computer science made with their as-yet unreleased model Astra. The discoveries cover a wide range of mathematical fields, including geometry, cryptography and coding theory. These ten are just the latest in a rapidly growing list of mathematical breakthroughs made by generative artificial in
As AI agents’ ability to do science autonomously continues to improve, scientists are facing a once-unthinkable question: will I get scooped by a bot? At least twice in the past five weeks, researchers have said that they had been working for an extended period on a particular research question, only to learn that artificial-intelligence companies had either answered the question before them or announced results befo
Black-box optimization (BBO) arises in many scientific and engineering problems where objective evaluations are expensive and limited. Recent large language model (LLM) agents offer a new way to approach BBO by combining task semantics, computation, optimization tools, and feedback-driven decision making, showing great potential due to the integration with mathematically rigorous tools. However, existing agentic BBO
A few days ago OpenAI released a paper generated by "an internal frontier model" of Navier-Stokes fame titled A Coulomb ground-state density without Kohn-Sham ensemble representation . The model considers a system with 2 nuclei and 3 electrons. It then spends 40 pages (seemingly) proving that for "sufficiently large" internuclear distance, restricted open-shell ensemble KS-DFT cannot describe the ground-state density
The Mathocalypse Last night my 9-year-old son was taunting my wife, complexity theorist Dana Moshkovitz, as follows: “mommy, I heard you got cooked! I heard that a robot solved the math problem you worked on for your whole career! OOF!” While my son was being a brat, he also wasn’t wrong. Whether you’re thrilled, depressed, angry, or whatever else about it, yesterday was surely one of the biggest days in mathematical
OpenAI’s massive new math drop: [This essay was written in extreme haste before a very long wifi-less flight; please forgive typos.] Part One: My take The real news here isn’t the result; it’s not what we were told. 1. AI once tried to be a science. Now we get stuff like the completely vague report from OpenAI below: “Same procedure”? “Using an unreleased model”? This would never pass peer review. We don’t know what
OpenAI releases 722 AI-generated mathematics papers as researchers weigh what comes next OpenAI has published hundreds of mathematical manuscripts produced by an unreleased AI model, opening a much bigger debate over how AI-generated research should be checked, understood and incorporated into mathematics OpenAI has released a collection of 722 mathematical manuscripts generated by an internal frontier AI model, cove
OpenAI has released 722 mathematics manuscripts produced by a single unreleased internal model, grouped into 372 results across 17 areas. According to the company, the model tried roughly 4,000 research problems, with each result consuming compute equal to about three hours of ChatGPT Pro thinking. Papers, source code and some Lean proofs were posted to GitHub, along with 10 model reasoning summaries. Headline claims
AI Models & Platforms OpenAI Releases 722 Math Manuscripts From an Unreleased AI Model Add Unite.AI to your preferred sources on Google In a research post published October 6, 2026, OpenAI said it is releasing a broad range of new mathematical results produced by an unreleased internal frontier model, publishing 722 manuscripts organized into 372 families in a public GitHub repository alongside Lean proof formalizati
The floodgates are open. Right on the heels of OpenAI’s new large language model producing the biggest math breakthrough in two decades, the company just released no less than 372 results from the same model. Each resolves or makes substantial progress on a major open question in mathematics or theoretical computer science, the company says. OpenAI revealed the results in a GitHub repository at 6 P.M. EDT. The deluge
[This is a guest post by Ben Antieau , cross-posted from his blog . — T.] There was no personal problem, no world problem, whose eloquent solution did not exist— somewhere in some hexagon. —Jorge Luis Borges, “The Library of Babel” [1] New repository I am writing to announce Hexagon , a new repository for research works in the mathematical sciences and theoretical computer science. This has been a joint effort of man
[This is a guest post by Thomas Bloom , crossposted from the Erdős problems forum . This blog post was initially written in a different file format and converted using AI. — T.] AI is changing everything, for better and/or worse, and the rate of change is dizzying; in recent months this has been particularly evident in mathematics, where AI has gone from being essentially useless to helping solve some of the hardest
Terence Tao used a SAIR talk to criticize AI model companies for accelerating mathematics without knowing what comes next, saying, "We must slow down." He described a normal proof lifecycle—proposing a solution, mechanical verification, exposition, peer review, and textbook incorporation—and said AI is racing through generation and Lean formalization while the later stages stall, creating "proof indigestion": proofs
Large language models (LLMs) have become increasingly capable problem solvers, but being able to solve a problem is not the same as being able to teach it. Existing approaches to training LLMs as teachers rely on demonstrations, preference data, or predefined pedagogical criteria that specify what good teaching looks like. However, these signals are often not grounded in individual student learning outcomes, where ef
On 13 September, leading luminaries and up-and-coming talents in mathematics and computer science congregated at the annual Heidelberg Laureate Forum in Germany for a week of discussion, networking, and—naturally—wurst. Having attended several of these events in the past, the mathematicians normally focus their chatter on which famed researchers are present, or what interesting problems they have been working on. But
A language model can give a correct answer more probability than any single incorrect answer and still usually sample an incorrect one, because the incorrect answers together hold more probability. The power distribution raises each complete answer's probability to a power above one and renormalizes, shifting probability toward answers the model finds most likely (sharpening). Sampling from it improves reasoning with
[This is a guest post by Diego Córdoba and Luis Martínez-Zoroa . This blog post was initially written in a different file format and converted using AI. — T.] Euler formulated the equations for an inviscid incompressible fluid more than 250 years ago. Published in his 1757 memoir , they belong to the earliest systems of partial differential equations in mathematical physics, following d’Alembert’s work on the wave eq
On-policy distillation (OPD) has emerged as a widely used paradigm for post-training large language models, reducing the train--test mismatch of conventional distillation by supervising the student on its own generated trajectories. However, existing OPD objectives remain largely token-local and outcome-agnostic, optimizing teacher--student agreement at each prefix despite reasoning quality being determined at the tr
Numerical Linear Algebra (NLA) has consistently played a vital role in advancing science by providing tools to solve fundamental problems encountered in scientific and engineering applications. Over the decades, it has continually evolved to meet the demands driven by successive waves of scientific discovery. For instance, during the 1950s and 1960s, substantial efforts were devoted to developing methods for solving
Solving Open Research Problems Together Mathematicians and Muse Spark collaborate on six research papers Earlier this year, our models achieved gold-medal-level performance across five high-school Olympiad competitions in mathematics, physics, and chemistry. Those results inspired us to see whether AI could also help scientists solve open research problems. Competition problems can be incredibly difficult, but those
Masked diffusion language models (dLMs) offer a promising parallel alternative to autoregressive models for complex reasoning. However, they face a distinct credit-assignment challenge, since a few commitments during denoising sharply reduce the uncertainty over the remaining masked positions and shape much of the response. Most post-training recipes for dLMs do not use this signal to decide which tokens to train on:
Noam Brown, an OpenAI researcher and core author of o1, discussed his current focus, multi-agent systems, in a recent podcast. He said OpenAI previously used 10,000 agents over 88 hours and 130 billion tokens to solve a Millennium Prize mathematics problem. Brown was explicit that the result came from having a general, powerful model, and that he would not even attribute 10% of the credit to multi-agent methods, whic
[This is a guest post by a coalition of Caltech mathematicians: the original organizers of Mathathon (Alvan Arulandu, Andrea Li, Avni Garg, Brian Zhao, Caiman Moreno-Earle, Sathvik Redrouthu), two coauthors of the Open Letter about the Mathathon (Dylan King, Tasmin Chu), and other members of the Caltech community (Mayla Ward, Merrick Hua, Robert Joseph George, Sergei Gukov, Shaowu Zhang, Tony Yue Yu, Vihaan Dheer). T
On-policy learning has been argued to reduce catastrophic forgetting, produce sparser parameter updates, and improve generalisation. However, existing comparisons between supervised fine-tuning and reinforcement learning vary many factors simultaneously, making the contribution of rollout policy difficult to isolate. We study the effect of rollout policy in a controlled strong-to-weak distillation setting, by indepen
Language models can now prove theorems, but people still decide which problems to pursue. We ask whether a model's internal representations can help identify promising mathematical connections. We develop LANTERN, a fast, cost-efficient pipeline that uses a classifier over pretrained-model activations to rank candidate relations, followed by staged filtering, hypothesis generation, executable verification, and analyt
[This is a guest post by Tapio Schneider . This blog post was initially written in a different file format and converted using AI. — T.] [This will be cross-posted on the CliMA blog .] The apparent proof of finite-time blow-up of the forced Navier-Stokes equation, announced by OpenAI on September 8, has brought into focus a debate about the role of AI in mathematics. The proof was produced with a system of some 10,00
Two weeks ago OpenAI claimed a solution to one of the biggest open problems in math—the Navier-Stokes problem—an achievement worth a $1-million prize from the Clay Mathematics Institute. The proof ignited a powder keg of concern over artificial intelligence companies’ race to disrupt the subject. But with the dust still far from settled, a different controversy is emerging: Did OpenAI even solve the right Navier-Stok
Last year I co-founded the Foundation for Science and AI Research (SAIR) with some private donors to create a non-profit organization that could support responsible uses of AI in mathematics and the other sciences, independent of the major AI companies. Initially, our resources were rather limited, and our main activities consisted primarily of podcasts , short events , and competitions . However, we have been slowly
OpenAI’s announcement that it used an artificial-intelligence model to solve a major puzzle in fluid dynamics — the Navier–Stokes problem — is likely to change how mathematics is done forever. The claim, made by the AI company in San Francisco, California, on 8 September, has also sparked controversy about whether OpenAI tools have learnt information from human mathematicians who were also racing to solve the problem
Large Reasoning Models (LRMs) achieve strong performance on complex tasks but exhibit systematic inefficiency: they often overthink easy problems and underthink hard ones. Existing approaches based on uniform length penalties or rigid routing incur an efficiency tax, trading reduced computation on easy instances for accuracy loss on hard instances. We formulate efficient reasoning as an instance-adaptive computation
On-policy self-distillation (OPSD) lets a language model learn from a frozen copy of itself that sees an answer or a worked solution. Giving the teacher this extra information seems to offer the student more to learn, but how much does it add beyond distillation itself? To isolate that contribution, we construct AMPLE-Math, a reusable suite of 5,319 mathematical problems with six reasoning views that share the same a
On September 8, OpenAI said 10,000 of its agents, running for 88 hours, had produced a 166-page proof regarding the Navier-Stokes equations — which describe fluid movement and have remained an open question since 1934. It's one of the seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000, each carrying a $1 million prize; only one has ever been awarded. OpenAI has said it won't be clai
A statement signed by 25 Fields medalists, posted by Terence Tao on September 11 under the title 《人工智能在数学中的严重错位》, argues that AI companies' focus on solving famous mathematical problems seriously diverges from the community's goals of conceptual understanding, method development, and knowledge transmission. Former OpenAI VP of post-training Liam Fedus responds that today's AI limits should not frame future possibilit
[This is a guest post by Silvia De Toffoli and Eamon Duede . This blog post was initially written in a different file format and converted using AI. — T.] Silvia De Toffoli (University School for Advanced Studies IUSS Pavia) Eamon Duede (Princeton University and Purdue University) On September 8th, 2026, OpenAI announced that it had produced an AI-generated solution to the Navier–Stokes existence and smoothness probl
It was a week that left mathematicians reeling. Hot on the heels of a flurry of cases of artificial intelligence furthering the field, OpenAI declared a major scalp: its latest AI model had cracked a Millennium Prize Problem, a puzzle with a $1m reward that had defied human brains for decades. The achievement bore little resemblance to how mathematical problems normally fall. A near-trillion dollar private company ha
Epoch AI says FrontierMath Tier 4 is now "saturated": by its accounting, when attempts across different models and different time periods are accumulated, every Tier 4 problem has been correctly solved by an AI at least once. OpenAI reports a score of 97.6% for GPT-6 Astra, which solved the one problem that had never been cracked before. When Tier 4 launched on July 11, 2025, the top score on the board was only about
Twenty-five Fields Medalists, with Terence Tao and Deng Yu among the first signers, issued an urgent public statement arguing that the way AI companies are pursuing mathematics is now in "severe misalignment" with the goals of the mathematical community. The statement explicitly does not call for stopping AI or deny its achievements. It raises three concerns: AI is solving problems faster than the community can diges
[This is a guest post by Lucas Fagan . This blog post was initially written in a different file format and converted using AI. — T.] I am excited to announce the Andrews–Curtis Conjecture Challenge , which opens today. This challenge is a collaboration between the SAIR Foundation and the Math-AI group at Caltech , organized by Sergei Gukov, Terence Tao, and myself. The Andrews–Curtis conjecture is one of the most pro
The controversy over OpenAI's maths miracle ‘Historic’ mathematical breakthrough by the tech company has been overshadowed by data privacy concerns of human researchers This year has been one of “unthinkable” progress for maths, said The Wall Street Journal. But OpenAI’s proof of the Navier-Stokes problem – one of the seven esteemed Millennium Prize Problems accepted as maths’ “deepest and most difficult unresolved q
Reinforcement learning (RL) has become central to post-training of large language models. Recent advances in RL for Mixture-of-Experts (MoE) models have primarily focused on improving optimization stability and training efficiency, while treating the expert selection as a fixed component. Since routing determines the sparse computation paths that induce output distributions, expert selection offers an additional sour
We demonstrate that training LLMs with RL does not improve performance equally across a dataset. RL shows large improvements on easy problems that an LLM is already good at solving, but small improvements on hard problems. We call this the Matthew Effect in RL for LLMs, after the phenomenon of cumulative advantage from economics and network science summarized as "the rich get richer". The naive explanation is that ha
Reinforcement learning with verifiable rewards is typically performed on-policy, keeping training data close to the current policy but limiting learning to trajectories that the policy can discover itself. Off-policy methods such as supervised fine-tuning, on the other hand, can leverage external knowledge beyond the base model's capabilities, but may suffer from large distribution shift. The key challenge is thus to
[This is a guest post by Anima Anandkumar . This blog post was initially written in a different file format and converted using AI. — T.] The last few days have been a flurry of activity. After months of sleepless nights and diving deep into different mathematical and computational tools to tackle the open problem of singularity in fluid dynamics, we were finally ready to send our work on Euler equations (without for
Neurosymbolic systems rely on mathematical solvers to guarantee reasoning correctness, yet solvers are fundamentally blind to whether a formal translation maintains strict reference-equivalence to a designated formalization. We formalize this vulnerability as Verdict-Preserving-Unfaithfulness (VPU): a failure mode where an incorrect encoding executes successfully and matches the expected verdict. We theoretically pro
Today was a tough news cycle to launch anything; we ordinarily promise to cover any new decacorn fundraises so Cognition’s $48B round and Mistral’s $24B round would normally have made it; we love imagegen so GPT Image 2.5 would have been its own headline; we covered the Dreamer story closely so their relaunch as Meta’s Muse agent should have made it; but.. yknow… the bar is higher these days. The summaries below capt
There’s some exciting very recent work by Alpöge and Buckmaster , building upon prior work by Córdoba and Martínez-Zoroa , in the general topic around the infamous global regularity problem for the incompressible three-dimensional Navier-Stokes equations . It is now widely expected that it should be possible to construct smooth initial data and smooth forcing term that would make these equations develop singularities
We investigate the best $L_2$ approximation of mixed Sobolev spaces by shallow neural networks with $n$ neurons and general activation functions. We first establish an activation-independent Fourier-block principle: if an activation has univariate approximation order $\rho$ in the sense of the Fourier-block property, then the global approximation rate has algebraic order $\min\{\alpha,\rho\}$ for target functions of
Respect where respect is due: Anthropic says Claude completed the first fully computer-checked proof of Fermat’s Last Theorem in 11 days. Andrew Wiles proved the theorem in 1995. Claude’s achievement was turning an existing proof into a form where a computer can check every… https://t.co/FcRbJn5GYb pic.twitter.com/OOdfoI2AZD
Another serious win for AI in mathematics: Claude formalized Fermat’s Last Theorem in 11 days. AI may now finally be able to automate the extremely labor-intensive job of turning advanced human mathematics into proofs that software can check line by line. The process took 11… https://t.co/DT5R8t4PbH pic.twitter.com/hRuE9pFCRo
Checking that a major mathematical proof is correct can take years. Formalization—converting the mathematical reasoning into a form computer proof assistants like Lean can verify—can help. Last month, Claude completed the first formalized proof of Fermat’s Last Theorem, one of… pic.twitter.com/pdT8zwlV4A
Science Sep 4, 2026 We are sharing the first complete computer-checked proof of Fermat’s Last Theorem. Claude worked largely autonomously over 11 days to write the proof in the Lean programming language. Below, we describe how the formalization was done and share some thoughts about what this work could mean for research mathematics. Around 1637, Pierre de Fermat jotted down a claim in the margin of his copy of Dioph
We report on our ongoing project to develop a computational pipeline, AutoGraphForge, for an automated graph-theoretic conjecturing-refuting-formalizing-proving system. Conjecture generation is counterexample-guided and runs in rounds: a Graffiti3 generator proposes conjectures over a small, evolving snapshot table $T$ (initially a few hundred graphs with their computed invariants) that grows only by counterexamples
Large language models (LLMs) achieve strong performance on mathematical reasoning benchmarks, yet the mathematically meaningful skills underlying their reasoning remain underexplored. We introduce \texttt{SHAPE}, a framework that analyzes Chain-of-Thought (CoT) trajectories through two lenses developed in mathematics education: (1) semantic spaces: the model's evolving mathematical interpretations of a problem (e.g.,
Network comparison using optimal transport is a growing area of research in network science. Unlike standard graph metrics, optimal transport computes both network dissimilarity and a transport plan that explains how one graph morphs into another. In this paper, we review how optimal transport compares undirected, unweighted graphs using three primary distances: the Wasserstein, Gromov-Wasserstein, and Bures-Wasserst
Physics-Informed Neural Networks (PINNs) have recently gained considerable attention as a mesh-free framework for solving partial differential equations. Nevertheless, their performance deteriorates when applied to strongly coupled multiphysics systems, such as Biot's consolidation model, due to severely ill-conditioned optimization landscapes. In this work, we propose a robust PINN-based solver, termed FS-ENGD-PINN,
Lyapunov's direct method is a cornerstone of stability and control, but it hinges on finding a Lyapunov function, a task demanding ingenuity or computation. A key difficulty is that every sub-level set of the function must be forward invariant, coupling its geometry to the system's trajectories. We relax this by replacing invariance with recurrence: a set is recurrent if every trajectory starting in it returns within
While Physics-Informed Neural Networks (PINNs) have emerged as a transformative paradigm for solving complex differential equations, their reliance on backpropagation-based gradient descent and automatic differentiation (AD) imposes significant computational bottlenecks and severe non-convex optimization challenges. To overcome these fundamental limitations, we propose the Physics-Informed Stochastic Configuration Ma
Physics-Informed Neural Networks (PINNs) have emerged as an important class of numerical methods for solving partial differential equations (PDEs). However, during the late-stage optimization process, further parameter updates often yield diminishing accuracy improvements while increasing computational costs. To address this issue, this paper proposes a Physics-Informed Error Field Learning (PIEFL) framework for PINN
Multilingual language models can solve the same mathematical problem in different languages, but it remains unclear whether they rely on shared features or on language-specific computations that only produce similar outputs. We study this question in five models from four families using the Multilingual Grade School Math (MGSM) dataset, with problems solved in English, German, French, Spanish, Russian, and Chinese, r
On-policy distillation trains a language model on its own generations while a teacher scores them token by token. It combines the dense supervision of imitation learning with the on-policy sampling of reinforcement learning. But it requires a second, larger model to act as teacher. On-Policy Self-Distillation (OPSD) removes that cost. The teacher is the model itself, conditioned on privileged information the student
AI systems increasingly generate alternatives, inspect evidence, and deploy a selected output. Validation is therefore target-relative: evidence certifies deployment only in directions resolved by the interventions that produced it. We represent validation and deployment rules as kernels over a reliability surface. Their span geometry separates replication, which reduces sampling noise, from new intervention directio
This paper studies the Sinkhorn distributionally robust hypothesis testing (SDRHT) problem, seeking a robust detector against least-favorable distributions in Sinkhorn discrepancy-based ambiguity sets centered at the empirical distributions. Existing approaches solve this problem by solving large-scale conic programs, which are not scalable. To overcome this, we propose a generative framework that learns least-favora
These lecture notes form the first part of a master's-level course on advanced numerical linear algebra. Their aim is not only to present the classical algorithms, but to show why the subject has become considerably more central than it was a generation ago. Numerical linear algebra grew up alongside the numerical solution of partial differential equations, and for a long time that is where its large sparse systems c
Fields Medalist Jacob Tsimerman announced at the award ceremony that he will soon join OpenAI to conduct AI safety research. Author and source: The Washington Post Jacob Tsimerman, who just won the Fields Medal last month, announced at the award ceremony that he will soon join OpenAI to work on AI safety research. In early August, he convened around 40 mathematicians at OpenAI’s headquarters in San Francisco with res
Introducing MathForm, an open-source framework, dataset, and model for mathematical autoformalization with Lean 4. Formalizing mathematics makes mathematical knowledge machine-checkable, but it is more than translating statements into code. A model must map each concept onto… (source truncated).
On this episode of Decoder, The Verge's London-based AI reporter Robert Hart discusses the impact of AI on the field of mathematics and the existential crisis many leading mathematicians are experiencing. OpenAI's recent release of solutions to longstanding math problems has caused a stir in the math community, sparking intense debate. While AI systems still struggle with elementary arithmetic, they are becoming incr
This study addresses data-driven prediction of coarse-grained dynamics in multiscale PDE systems. Adopting a closure-free operator-learning viewpoint, the authors apply a linear coarse-graining map and learn a surrogate evolution operator for the resolved field directly from filtered high-fidelity trajectories. Motivated by the Mori-Zwanzig formalism, they propose a spatiotemporal neural operator mapping a resolved h
In this episode of AI Pioneers, Cynthia Rudin, a pioneer in interpretable machine learning and winner of the 2022 Squirrel AI Award, discusses her work on black boxes and Rashomon sets. She recounts her early project with Con Edison on the NYC power grid, where black-box models failed and simple interpretable models outperformed them. Rudin argues that for high-stakes problems like loan decisions or crime prediction,
Hyra continues to push the frontier. Since our last math update, four more advances have landed: In 3D Blaschke–Lebesgue, the universal lower bound for constant-width bodies has been raised from 0.380799w³ to 0.411040w³, now reaching 97.9% of the conjectured Meissner optimum. Additional advances include Beurling–Ahlfors and others.
In recent months there has been a proliferation of AI-generated proofs of various old and new results, some of which have been formalized in the proof assistant language Lean. However, checking that a given Lean repository actually proves the claimed statement is somewhat non-trivial, especially for an audience which is not expert in the use of Lean: one has to first check that the claimed formal Lean statements have
This paper introduces Chain-of-Experience (CoE), a paradigm enabling large language models (LLMs) to improve at test time through iterative interaction with self or environmental feedback. Across math, coding, and knowledge domains, using 8 LLMs including GPT-5, Gemini-2.5 Pro, and Claude-4.5 Sonnet, the authors show that leveraging iterative experience consistently outperforms feedback-free baselines, with self-feed
The human brain exhibits a striking degree of functional specialization, with distinct networks supporting language, formal reasoning, reasoning about other minds, and reasoning about the physical world. Is this modular organization a fundamental principle of how intelligent systems must be built, or an evolutionary accident specific to biological brains? Here, we test whether a similar organization emerges in Large
Frontier LLMs are updated frequently and typically outperform their predecessors in aggregate. But aggregate gains say little about individual samples: an update can still cause sample-level regression, where a response correct under the old model becomes incorrect under the new one. This paper studies how to predict such regressions from signals available at inference time. We compare single-model signals (confidenc
In offline reinforcement learning, estimating the optimal action-value function Q* can be formulated as solving the optimal Bellman equation based solely on offline observations. A fundamental challenge is that the reward function and transition kernel are unknown, so the optimal Bellman operator is not directly observable. To address this, we propose a novel framework that decouples operator estimation from value fu
AI systems are increasingly contributing to mathematical research, but frontier-model reasoning and expert review are scarce resources that must be allocated efficiently. Current AI-for-math workflows concentrate human effort on problem selection and artifact review, which are becoming bottlenecks. To address this, the authors propose a new human-AI discovery paradigm in which humans specify a research direction, and
MathCode converts natural-language mathematics problems into Lean 4 theorem statements and attempts formal proofs. Its public project combines a persistent Lean REPL, reusable theorem and axiom libraries, agentic proving, parallel subgoal trees and an Obsidian theorem graph.
Autoformalization is commonly framed as translating natural-language mathematical statements into machine-verifiable formal languages such as Lean 4. However, faithful formalization requires more than translation; models must map concepts to the complex hierarchy of types and definitions in formal libraries like Mathlib. Existing approaches struggle because they rely heavily on parametric memory and lack feedback-dri
DeepMind’s Pushmeet Kohli on AI’s Scientific Revolution Pushmeet Kohli leads AI for Science at DeepMind, where his team has created AlphaEvolve, an AI system that discovers entirely new algorithms and proves mathematical results that have eluded researchers for decades. From improving 50-year-old matrix multiplication algorithms to generating interpretable code for complex problems like data center scheduling, AlphaE
This post concerns the following conjecture of Sendov , as well as its strengthening by Phelps–Rodriguez : Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with . Conjecture 2 (Phelps–Rodriguez conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , the
At a glance MindTopo is a new benchmark for testing topological reasoning in AI, evaluating whether multimodal models can understand concepts such as connectivity, enclosure, order, separation, and knots. The benchmark measures both reasoning and planning, testing not only whether models can recognize topological relationships in static images but also whether they can preserve and manipulate those relationships thro
For the sake of anyone who might read this blog post in the distant future (a month from now, say), let me mention that I am writing it a few days after OpenAI announced that it had solved ten major problems in mathematics and theoretical computer science, including the first construction of a non-sofic group, and a proof that the multicolour Ramsey number (where there are 3’s) grows superexponentially in . The first
Science Aug 10, 2026 Recently, a member of staff at Anthropic gave Claude an unreasonable challenge. It was about one of the most famous unsolved problems in mathematics: Take a real stab at the Riemann hypothesis. Claude did take a real stab, but as you might have expected if you’re familiar with the difficulty of the task (the Riemann hypothesis dates back to 1859 and has a million-dollar bounty), it didn’t succeed
Terence Tao discusses the Heil-Ramanathan-Topiwala (HRT) conjecture, which asserts that a nonzero function with reasonable decay cannot satisfy a finite linear relation among its time-frequency shifts. He notes that the conjecture holds when all shifts are frequency shifts or collinear. For the lattice case, commutativity allows a diagonalization that reduces the relation to a pointwise equation, using restrictions t
How OpenAI Announced Its Next Frontier Model 'Astra' The US AI firm OpenAI, led by CEO Sam Altman, has announced its latest model, Astra. In a shocking introduction to the AI model, the company shared a selection of 10 results, each of which resolves or makes substantial progress on a long-standing open problem. The problems span high-dimensional geometry, coding theory, arithmetic circuit complexity, group theory, o
OpenAI says an internal version of its next major artificial intelligence model has produced new results on 10 longstanding problems in mathematics and theoretical computer science, highlighting the growing role of AI systems in advanced scientific research. The company said the results were generated by an internal version of Astra, described as its next major model. The issues had remained open with no progress on
OpenAI released ten results on long-standing problems together with human-prepared manuscripts and Lean certificates.
Tao, 51, an Australian and American mathematician recognized for his contributions to partial differential equations, combinatorics, harmonic analysis, and additive number theory, made the remarks in his lecture, "Mathematics in the Age of AI," delivered at the International Congress of Mathematicians (ICM) last weekend. His speech drew widespread attention from the global mathematics community. He began by reflectin
The author recounts attending a workshop on mathematics and AI, which produced the Leiden Declaration, now signed by over 3000 people. Although he did not sign, he clarifies it is not due to disagreement but rather uncertainty about some confident assertions and recommendations. He appreciates the declaration's role in focusing attention on pressing issues, and quotes its passage on characteristic values of mathemati
A Belief Flow Network (BFN) is a directed graph where nodes represent agents holding epistemic states—complex logical beliefs including conditional statements. Agents exchange information asynchronously, revising their beliefs via iterated belief change operators, potentially with different rational policies (some skeptical, some credulous). The research asks whether consensus is inevitable if influence can eventuall
The notorious Jacobian conjecture can be formulated concretely over the complex numbers as follows. Conjecture 1 (Jacobian Conjecture) Let be a polynomial map in complex variables, whose Jacobian is a non-zero constant. Then is invertible (with polynomial inverse). The condition that the Jacobian is non-zero is equivalent to being locally invertible. (The implication of local invertibility from non-vanishing Jacobian
It’s been an interesting few weeks for counterexamples. This post is basically my perspective of what has been going on in the world of formalization, AI tools and, in particular, counterexamples. Unit distance Two months ago today (20th May 2026), ChatGPT disproved Erdős’ Unit Distance conjecture in discrete geometry. This is now old news but I had to start somewhere. The announcement was accompanied with testimonie
Terry Tao demonstrates how coding agents facilitate converting code-generated figures into interactive applets, using Figure 1 from his recent paper on the Gilbreath conjecture with Chase and Hunter. The figure plots exact and Monte Carlo simulated values of a sequence called the "Gilbreath expectation sequence," which arises from a Gilbreath array: an inverted pyramid with top entries as independent exponential rand
I am finding the newly revealed capability to code old applet ideas into reality to be very tempting to sink more time into, though I am certainly encountering the common “vibe coding” experience that the process can produce something that superficially resembles a finished product well before a satisfactory level of testing and review has been completed; indeed, it is the review process which is now the most time-co
I have been interested in machine-assisted ways to do and teach mathematics from as far back as 1999, when I started coding several applets in Java 1.0 , both for my complex analysis and linear algebra courses, as well as to visualize various mathematical objects I was interested in (such as honeycombs or Besicovitch sets). This was moderately successful; but the applets were time-consuming to program. Eventually, th
In the latest interview with AAAI/SIGAI Doctoral Consortium participants, Thi Kieu Khanh Ho, a PhD student at McGill University and Mila – Québec AI Institute, discusses her research on time-series anomaly detection. Her work focuses on teaching AI systems to detect unusual events in complex real-world data streams without relying on large amounts of labeled examples, which are often scarce, noisy, or contaminated. S
Last week, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, a famous problem in discrete geometry that had stumped human mathematicians for the last 80 years. OpenAI gave several mathematicians early access to the result and published their reactions. Tim Gowers — who won the Fields Medal, the most prestigious prize in mathematics — wrote that “there is no doubt that the so
OpenAI scored a big result yesterday, with respect to an Erdos problem: Clearly impressive. But as with so much else, it should be viewed with skepticism. In an email to me this morning, Cal Newport made a number of good points that he said I could share, summarizing both what was found and some limitations: OpenAI used a new reasoning model (not yet released) to help identify a counterexample that disproved a conjec
I’m organizing a workshop in London on July 6th to 10th (2026) whose goal is to work on my EPSRC-funded project formalizing Fermat’s Last theorem in Lean. The initial aim of the project was to reduce FLT to theorems known in the 1980s, i.e. “formalize the Wiles/Taylor–Wiles papers” (although actually we are formalizing a more modern proof whose ideas go back to a strategy proposed by Khare). I have just finished givi
We are all having to keep revising upwards our assessments of the mathematical capabilities of large language models. I have just made a fairly large revision as a result of ChatGPT 5.5 Pro, to which I am fortunate to have been given access, producing a piece of PhD-level research in an hour or so, with no serious mathematical input from me. The background is that, as has been widely reported, LLMs are now capable of
Google DeepMind presents an agentic system designed to collaborate with research mathematicians on open-ended problems.
Terence Tao argues that generating and verifying proofs is still incomplete until results are digested and connected to the field.
In a Nature interview, Tao separates current research usefulness from hype and explains the growing role of formal verification and human judgment.
Jùn Zhang and Carina Hong discuss translating mathematics into Lean, formal proof, mathematical intuition and building an AI-for-math company.
UCLA Computer Scientists, Mathematician Terence Tao Awarded Laude Moonshots Grant to Build AI System as Research Partner Project aims to unlock progress in physics, cryptography and other challenging fields The multidisciplinary UCLA team will lead the Accelerating the Queen of Sciences project to develop an AI system capable of thinking like a mathematician and accelerating scientific breakthroughs. A team of six UC
Tanya Klowden and Terence Tao examine how AI and formalization may change the purpose, practice and social role of mathematics.
"AI Has Overwhelmed Human Exam Questions," A Conversation Between Terence Tao and Mark Chen: Large Language Models Are No Longer "Math Underachievers" Recently, at a seminar hosted by the Institute for Pure and Applied Mathematics (IPAM), Fields Medalist Terence Tao and Mark Chen, the Chief Researcher at OpenAI, participated in a "fireside chat." The two discussed the leap - forward progress of AI in the past year an
Much of the excitement around the news has stemmed from the adjudicator of these AI-written proofs: Terence Tao, a UCLA professor who is widely considered to be the world’s greatest living mathematician, reported The Atlantic. When Tao was interviewed about what AI can offer mathematics, he was more tempered. The AI-generated Erdős solutions are impressive, he said, but not overwhelmingly so: The bots have functional
Accelerating Mathematical and Scientific Discovery with Gemini Deep Think Under direction from expert mathematicians and scientists, Gemini Deep Think is solving professional research problems across mathematics, physics, and computer science In the summer of 2025, an advanced version of Gemini Deep Think achieved Gold-medal standard at the International Mathematics Olympiad (IMO) and later, an updated version, obtai
Let’s say that someone had a big pot of money, and wanted to use it to accelerate mathematical discovery. How might they go about doing this? The traditional approach Historically it has been governments who have been driving this agenda, with taxpayer-funded institutions such as the NSF (in the US), the EPSRC (in the UK), the ERC (in Europe) and many other national institutions (NSERC, ARC, DFG, JSPS…) typically fun
Epoch AI released a pilot of 14 research-level open problems with programmatically checkable answers to test the frontier beyond existing benchmarks.
[This is a guest post by Boris Alexeev . Now over to Boris.] I’m here to tell you about various exciting developments centering on Erdős problems, especially involving the formalization of old and new mathematics using artificial intelligence. Background As is well known, Paul Erdős was a prolific mathematician of the 20th century who posed an extraordinary number of conjectures. Around May 2023 , Thomas Bloom set up
Google DeepMind’s AlphaEvolve AI is accelerating mathematical research at an unprecedented scale, according to a new paper published this week by collaborators including renowned mathematician Terence Tao. The research shows how the AI agent tackled 67 challenging problems, rediscovering top solutions and finding novel constructions for several long-standing challenges. The work demonstrates a powerful new method for
So it’s an interesting time for computers-doing-mathematics. A couple of interesting things happened in the last few days, which have inspired me to write about the question more broadly. First there is the question on whether computers will ever prove interesting research-level mathematical theorems autonomously at all, which is really hard to answer right now, although clearly a lot of people are betting on “yes”.