Is AI the End of Math As We Know It? You bet!

Any doubts! What computers have not been able to accomplish for several decades, AI will advance Math faster and further!

Is AI the End of Math As We Know It? | Quanta Magazine

Sharing AI progress in mathematics by released more than 700 AI-generated papers | OpenAI

This is only the beginning! What an avalanche! Breathtaking!

How fast will mathematics (the queen of the sciences according to Carl Friedrich Gauss) now advance going forward? Maybe in the coming months/quarters we will see more progress than in the past 1000 years or so.

"OpenAI released more than 700 AI-generated papers presenting results related to more than 370 previously unsolved mathematical problems, including some of the field’s hardest. The massive dump marks yet another landmark in the use of AI to attack complex math—but experts are split over whether such releases are good for the field. Learn about what’s in the papers—and how researchers are reacting."

Sharing AI progress in mathematics | OpenAI

OpenAI's math solutions aren't meeting the field's standards yet | TechCrunch

Watch: What’s the deal with OpenAI’s massive dump of math papers?

Twenty-five leading mathematicians signed an open letter arguing that AI companies are threatening their intellectual work

That is almost unheard of! They say mathematics is the queen of all sciences (according to Carl Friedrich Gauss)!

Maybe machine learning and AI are dethroning math to a regular science! Maybe the once elite club of mathematicians ("mathematical community") lost its nimbus!

"Twenty-five leading mathematicians signed an open letter arguing that AI labs are threatening their intellectual work as they seek to one-up each other with solutions to famous math problems. Each signatory has been awarded the Fields Medal, considered the most prestigious prize in mathematics. ..."

"Over the last few months, the mathematical capabilities of LLMs have improved dramatically, to the point that they can solve major outstanding problems in many fields of mathematics. However, the push by AI companies to solve mathematical problems as a benchmark is detrimental to the science of mathematics, and to the mathematical community. The goals of the AI companies and the goals of the mathematical community are severely misaligned. We see these as part of broader alignment issues impacting other scientific and creative professions, as well as the whole of society. ...

The mathematical community functions, in many ways, as a miniature version of humanity. It consists of individuals using a wide variety of different approaches, joined by core values. The most precious resources of our profession are students and ideas, and these we nurture with great care. ..."

OpenAI's feud with mathematicians is only escalating | TechCrunch

A Severe Misalignment of AI in Mathematics

On the Navier–Stokes Millennium Prize Problem | OpenAI

Amazing stuff! Exciting times for math!

However, OpenAI fought dirty on career-making math problem, says NYU mathematician. "OpenAI fought dirty on career-making math problem, says NYU mathematician: There is a $1 million bounty for the first person providing a solution to the Navier-Stokes existence and smoothness problem." OpenAI admits there was apparently an intense competition going in the last few days or so on who would come out first.

"We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean. ...

The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra. ...

In the process of resolving the Navier–Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens. ..."

On the Navier–Stokes Millennium Prize Problem | OpenAI

Credits: The Wall Street Journal What's news

A snapshot of local incompressible motion. Orange marks faster angular rotation; teal marks slower rotation. Circulating speed also depends on radius. The trajectories show inward spiraling and axial stretching.

Huge Breakthrough in the Math of Imbalance

Recommendable! It took 15 years to refine the algorithm.

"... You can always make [two or more] teams [of extremely different individuals] surprisingly even, according to researchers studying combinatorial discrepancy theory.

Discrepancy theory is a branch of mathematics concerned with allocating resources as evenly as possible. ...

In the early 1980s, the mathematician János Komlós came up with a counterintuitive [conjecture]. He [predicted] that no matter how many objects (your players) or dimensions (... categories [or dimensions, attributes]) you consider, the discrepancy — which you can quantify — will never exceed a constant amount. There will always be a way to divide the teams with a discrepancy below that exact amount. ...

If the Komlós conjecture is true, it could unlock answers to many other problems, both within discrepancy theory and in fields like operations research. ...

Then, in fall 2025, [researchers] announced the first major advance on the problem(opens a new tab) in nearly 30 years. They found a limit that changes so slowly with the dimension that it is only a hair away from constant, even with an astronomical number of dimensions. ...

The Komlós conjecture imagines each person (or object) as an arrow of length 1 called a unit vector. This vector is defined by a list of coordinates, where each coordinate measures how much of a particular attribute that person has. ...

Now assign each vector to a team. If you put a vector in Team A, leave its coordinates alone. If you put it in Team B, multiply each of its coordinates by −1. (This flips the vector around.) ...

If you’re able to make a perfect split, dividing people into two teams so that each team has an equal ..., then all of these vectors should add up to zero. Perfect harmony.

But perfection usually isn’t possible. So the question becomes: How close to zero can you get? ...

Consider one naïve strategy: Simply assign vectors to teams at random. This leads to a discrepancy that skyrockets as the number of vectors, N, increases.

In 1985, Joel Spencer found a better bound, capping discrepancy below the logarithm of N;

in 1998, Wojciech Banaszczyk improved the bound to

, which can also be written as log(N)½. Both were meaningful strides, but the amount of imbalance still grew as the number of vectors did. Komlós’ constant felt out of reach. ...

In 2010, he came up with an idea for an algorithm(opens a new tab). He started by splitting each vector in half. ..."

‘Huge Breakthrough’ in the Math of Imbalance | Quanta Magazine "For the first time in 30 years, computer scientists have found a better way to allocate objects evenly between two groups."

Decoupling via Affine Spectral-Independence: Beck-Fiala and Komlós Bounds Beyond Banaszczyk (preprint, open access)

From the abstract

Graduate Student Proves the Fractal Uncertainty Principle

Recommendable! Until now I was only familiar with the Heisenberg Uncertainty/indeterminacy Principle in physics.

From the abstract (very short):

"We prove that if a fractal set in R^d avoids lines in a certain quantitative sense, which we call line porosity, then it has a fractal uncertainty principle. The main ingredient is a new higher dimensional Beurling-Malliavin multiplier theorem."

Graduate Student Proves the Fractal Uncertainty Principle | Quanta Magazine "The math, which combines chaos, quantum theory, and infinitely complex fractal structures, has been called a “foundational result.”"

Fractal uncertainty in higher dimensions (preprint, open access)

Christopher A. Sims (1942–2026): Paradigm-shifting macroeconomist and econometrician

R.I.P.

That makes 2026 the year renown econometricians died. This is the second one! See also my blog post here.

From the abstract:

"Christopher A. Sims, who died on March 14, 2026, left behind an indelible mark on macroeconomics and econometrics. His Nobel Prize winning research reshaped the paradigm for empirical macroeconomics, creating a rigorous and flexible framework for inferring the effects of macroeconomic policies. He also made important contributions to time series forecasting, Bayesian econometrics, factor modeling, approximation theory, and models of monetary and fiscal policy.

Few academics have had as direct an impact on the work of central banks throughout the world. A frank and generous mentor with numerous students, Chris’s impact extends well beyond his published papers."

Christopher A. Sims (1942–2026): Paradigm-shifting macroeconomist and econometrician | PNAS (open access)

Sims teaching a class in 2011 right after being awarded the Nobel Prize in economics.

Ten advances in mathematics and theoretical computer science | OpenAI

Amazing stuff! This is only the beginning!

"OpenAI released solutions to ten open problems in mathematics and theoretical computer science, generated by Astra, an internal unreleased model.

The problems span sphere packing, coding theory, group theory, and quantum complexity—each vetted by formal proofs in Lean.

Generating all ten solutions cost roughly $2,000 in compute at current API rates. The results include a disproof of Connes’s rigidity conjecture, new bounds on sphere-packing density, and a construction proving the existence of non-sofic groups. OpenAI framed the release carefully, noting that humans prepared the manuscripts but the mathematical arguments came from the system, a deliberate stance on authorship and attribution as AI systems edge into research collaboration."

From the abstract:

"We present a collection of results obtained by an internal OpenAI model, spanning mathematics and theoretical computer science:

1. High-dimensional sphere packing. The asymptotic strength of the Cohn–Elkies linear program is determined exactly. This gives an improved general packing bound in high dimensions and settles the corresponding Fourier sign-uncertainty problem asymptotically.

2. Binary and spherical codes. Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters. The spherical construction also recovers the sphere-packing exponent of Chapter 1.

3. Non-sofic groups. An explicit non-sofic group is constructed, resolving the question of whether every countable group admits finite permutation approximations. The argument uses property-(T) expanders and the binary Leavitt algebra.

4. Connes’s rigidity conjecture. Infinitely many pairwise nonisomorphic property-(T) groups are constructed with the same group von Neumann algebra, disproving Connes’s conjecture and answering a related finite-to-one question of Popa.

5. Arithmetic circuit complexity. For the permanent, division-free circuits require Ω(n2 log log n) gates, while formulas require Ω(n 4/ log n) leaves.

6. Quantum parallel repetition. Exponential parallel repetition is proved for every finite two-player entangled game, extending the classical repetition principle beyond previously treated special classes of quantum games.

7. Closest vector problem. A direct reduction from 3SAT gives n

1/400-factor hardness for the Euclidean closest vector problem, with related consequences for binary decoding and other lattice norms.

8. Ehrhart’s volume conjecture. The sharp bound (n + 1)n/n! is proved in every dimension for convex bodies whose barycenter is their only interior lattice point. 9. Multicolor Ramsey numbers. A superexponential lower bound proves Rk(3) = kΘ(k).

10. Compactness and degeneracy. Separate bipartite graph constructions disprove two conjectures in extremal graph theory: the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős."

Data Points newsletter

Ten advances in mathematics and theoretical computer science | OpenAI

Ten Advances in Mathematics and Theoretical Computer Science (open access, not peer reviewed, 249 pages)

Econometrician Takeshi Amemiya has died

R.I.P.

Yes, I remember Takeshi Amemiya! When I studied economics, I had a hard time with econometrics, time series analysis, and statistics since applied math is not my strong suit.

"... His work defined modern econometrics, which uses statistical methods and mathematical modeling to inform economic theory.

Amemiya’s body of scholarship was shaped by curiosity, discipline, and seriousness. His 1985 book, Advanced Econometrics, became an authoritative reference text and is still required reading at universities around the world. He worked on estimation of nonlinear econometric models including those for qualitative response, censored and truncated dependent variables, and transformed regressions. These models turned into standard tools across empirical economic fields. ..."

Econometrician Takeshi Amemiya has died | Stanford Report "One of the most influential econometricians of the last century, the Stanford professor harnessed statistical models to understand economic data, transforming the field of theoretical econometrics."

The Mind-Bending Physics of Origami

Very recommendable!

(440) The Mind-Bending Physics of Origami | NOVA | PBS - YouTube

Fields Medal win: Wang Hong, Deng Yu are first Chinese nationals to score top maths honour

Update of 7/29/2026: Yu Deng ’11 and Hong Wang PhD ’19 awarded Fields Medal "MIT-trained mathematicians earn the honor, one of the most prestigious in the field, for their significant achievements."

Congratulations! China, a science superpower! Competition is good, more competition is better! This could be a milestone!

"For the first time, China has produced Fields Medal winners educated on its own soil – and two at once."

"Yu Deng

For his work in partial differential equations, including the rigorous derivation of the Boltzmann equation from hard-sphere dynamics for rarefied gases, the derivation of wave kinetic equations from nonlinear dispersive systems, and probabilistic approaches to nonlinear Schrodinger dynamics."

Hong Wang

For her work in harmonic analysis and geometric measure theory, including applications of multiscale and decoupling techniques to the local smoothing conjecture for the planar wave equation, and major advances in Fourier restriction, Falconer distance sets, Furstenberg sets in the plane, and the Kakeya problem in three dimensions."

Developing | Fields Medal win: Wang Hong, Deng Yu are first Chinese nationals to score top maths honour | South China Morning Post (behind paywall)

Living Fully in the Math World Means Threading the Needle "Focus, commitment, and insight resulted in a “once in a century” proof. Hong Wang is now just the third woman to win a Fields Medal" (Caveat: I did not read the article)

Amid Life’s Chaos, a Meticulous Mathematician Finds Stability "Before he won a Fields Medal for his pioneering work balancing randomness and order, Yu Deng had to achieve equilibrium in his own life." (Caveat: I did not read the article)

IMU: Fields Medals 2026

International Mathematical Union: Fields Medal

Yu Deng & Hong Wang

Fields Medal

In game theory, generalists sometimes win out over specialists in reinforcement learning

This could be an interesting new paper by Zico Kolter.

"... it does offer new insights into so-called imperfect-information games that involve two contestants facing off in a “zero-sum” competition, where one player’s gain means the other player’s loss. ...

The focus of the new work is on algorithms that could be used to train neural networks to participate in imperfect-information games. The assumption, long-held in the field, was that algorithms grounded in principles of game theory would, in this setting, clearly outcompete a general-purpose variety of algorithms called policy gradient methods, which came into use for decision-making in the 1990s. The term “policy” in this context basically means strategy, whereas “gradient” refers to a path that leads in the direction of greatest change — to the top (or bottom) of a hill, for example. Policy gradient methods are being used to train neural networks to make decisions that move — in small, sequential steps — toward a particular goal (like reaching a summit, metaphorically speaking), with continual adjustments and course corrections made along the way to bring the agent closer to the intended destination. ..."

From the abstract:

"In the past decade, motivated by the putative failure of naive self-play deep reinforcement learning (DRL) in adversarial imperfect-information games, researchers have developed numerous DRL algorithms based on fictitious play (FP), double oracle (DO), and counterfactual regret minimization (CFR).

In light of recent results of the magnetic mirror descent algorithm, we hypothesize that simpler generic policy gradient methods like PPO are competitive with or superior to these FP-, DO-, and CFR-based DRL approaches. To facilitate the resolution of this hypothesis, we implement and release the first broadly accessible exact exploitability computations for five large games.

Using these games, we conduct the largest-ever exploitability comparison of DRL algorithms for imperfect-information games. Over 7000 training runs, we find that FP-, DO-, and CFR-based approaches fail to outperform generic policy gradient methods."

In game theory, generalists sometimes win out over specialists | MIT News | Massachusetts Institute of Technology "Researchers show that for certain kinds of games, an overlooked class of algorithms performs much better than expected."

Reevaluating Policy Gradient Methods for Imperfect-Information Games

On LEAP: Supercharging LLMs for Formal Mathematics with Agentic Frameworks

Google with Quoc V Le and hist team is bringing mathematics into the age of AI! Expect a lot of progress regarding mathematics in coming years thanks to ML & AI!

Exciting times!

From the abstract:

"Large Language Models (LLMs) exhibit strong informal mathematical reasoning but struggle to generate mechanically verifiable proofs in formal languages like Lean.

We present LEAP, an agentic framework that enables general-purpose foundation models to achieve state-of-the-art performance on automated formal theorem proving.

LEAP leverages foundation model capabilities, such as informal reasoning, instruction following, and iterative self-refinement. By decomposing complex problems into smaller units, the system bridges formal proof construction with informal blueprints through continuous interaction with the Lean compiler.

To provide a rigorous evaluation beyond increasingly saturated benchmarks, we introduce Lean-IMO-Bench, a benchmark of IMO-style problems formalized in Lean, with short statements yet highly non-routine and multi-step proofs across a wide range of difficulty levels.

Empirically, on the latest 2025 Putnam Competition, an annual mathematics competition for undergraduate students in North America, LEAP solves all 12 problems, matching recent breakthroughs by frontier formal mathematical models. On Lean-IMO-Bench, LEAP boosts the one-shot formal solve rate of general-purpose LLMs from below 10% to 70%, notably surpassing the 48% benchmark set by a specialized, gold-medal-caliber IMO system.

Furthermore, we demonstrate LEAP's research-level utility by autonomously formalizing complex proofs for open combinatorial challenges, including a verified proof for a key subproblem in Knuth's Hamiltonian decomposition of even-order Cayley graphs."

[2606.03303] LEAP: Supercharging LLMs for Formal Mathematics with Agentic Frameworks

With over 1600 signatories, mathematicians issue a declaration warning of AI. Really!

I bet, some mathematicians were also complaining when the Abacus was introduced more than 4,000 years ago! 😊 Same old story!

Of course, machine learning and AI is coming to mathematics, the queen of sciences (according to a famous quote from Carl Friedrich Gauss). I am already curious what ML & AI will contribute to mathematics. Apparently, it has already provided some proofs to some very old mathematical problems.

It is probably safe to largely ignore these warnings!

"Few communities have felt that pressure more acutely than mathematicians, who have haplessly watched AI get frighteningly smart, frighteningly fast.

Today, 16 math specialists have turned that unease into a public cry for help—and call to action. Part warning, part manifesto, the 11-page Leiden Declaration on Artificial Intelligence and Mathematics cautions that unchecked automation threatens not only how math is practiced, but what the discipline stands for. It also lays out principles for using AI in ways that support, rather than erode, the field. ..."

Mathematicians issue warning as AI rapidly gains ground | Science | AAAS

Leiden Declaration on Artificial Intelligence and Mathematics "This declaration calls for action to address the challenges posed by the use of artificial intelligence within mathematics research. It is the result of a community initiative and is endorsed by the International Mathematical Union (IMU)."

OpenAI model finds proof resolving famous mathematics problem dating from 1946

Amazing stuff! More to come! This is only the beginning!

"OpenAI model finds proof resolving famous mathematics problem

An OpenAI reasoning model has resolved the planar unit distance problem, a central question in discrete geometry posed by legendary mathematician Paul Erdős in 1946.

The conjecture held that square grid constructions were essentially optimal for maximizing unit-distance pairs among points in a plane, a belief that stood unchallenged for nearly 80 years.

The model instead found an infinite family of configurations yielding polynomial improvements over the grid approach, disproving the assumption.

What makes the breakthrough unusual is not just the result itself, but how it was found: a general-purpose reasoning model, not a system specialized for mathematics, produced a proof that external mathematicians have verified. The proof brings sophisticated tools from algebraic number theory to bear on an elementary geometric question, revealing unexpected connections between distant mathematical domains. Fields medalist Tim Gowers called it “a milestone in AI mathematics,” while number theorist Arul Shankar argued the result shows AI models “are capable of having original ingenious ideas, and then carrying them out to fruition.”" (Source)

An OpenAI model has disproved a central conjecture in discrete geometry

Paul Erdos (Source)

Previously known construction of many unit distances from a rescaled square grid.

A short biography of mathematician Georg Cantor

Recommendable!

The Man Who Stole Infinity | Quanta Magazine "In an 1874 paper, Georg Cantor proved that there are different sizes of infinity and changed math forever. A trove of newly unearthed letters shows that it was also an act of plagiarism."

Georg Cantor

On AI Co-Mathematician: Accelerating Mathematicians with Agentic AI

This sounds promising! I bet ML & AI will revolutise how we do math! This is only the beginning!

I wish I had a co-mathematician in high school! 😊

From the abstract:

"We introduce the AI co-mathematician, a workbench for mathematicians to interactively leverage AI agents to pursue open-ended research.

The AI co-mathematician is optimized to provide holistic support for the exploratory and iterative reality of mathematical workflows, including ideation, literature search, computational exploration, theorem proving and theory building. By providing an asynchronous, stateful workspace that manages uncertainty, refines user intent, tracks failed hypotheses, and outputs native mathematical artifacts, the system mirrors human collaborative workflows.

In early tests, the AI co-mathematician helped researchers solve open problems, identify new research directions, and uncover overlooked literature references.

Besides demonstrating a highly interactive paradigm for AI-assisted mathematical discovery, the AI co-mathematician also achieves state of the art results on hard problem-solving benchmarks, including scoring 48% on FrontierMath Tier 4, a new high score among all AI systems evaluated."

[2605.06651] AI Co-Mathematician: Accelerating Mathematicians with Agentic AI

Credits: Thinking Machines debuts a new type of model

Pure classical physics can explain quantum phenomena, study shows

Amazing stuff!

"A reformulation of the classical Hamilton-Jacobi equation, incorporating density and multiple least-action paths, can exactly reproduce quantum phenomena such as the double-slit experiment, quantum tunneling, and hydrogen atom wave functions. This approach mathematically bridges classical and quantum mechanics, showing that quantum behavior can be computed using classical principles without approximations."

" ... MIT scientists have now shown that certain mathematical ideas from everyday classical physics can be used to describe the often weird and nonintuitive behavior that occurs at the quantum, subatomic scale.

In a paper appearing today in the journal Proceedings of the Royal Society A Mathematical Physical and Engineering Science, the team shows that the motion of a quantum object can be calculated by applying an idea from classical physics known as "least action." With their new formulation, they show they can arrive at exactly the same solution as the Schrödinger equation—the main description of quantum mechanics—for a number of textbook quantum-mechanical scenarios, including the double-slit experiment and quantum tunneling. ..."

From the abstract:

"We show that the Schrödinger equation can be solved exactly based only on classical least action.

Fundamental postulates of quantum mechanics can in turn be derived directly from this construction. The results extend to the relativistic Klein-Gordon, Pauli, and Dirac equations, and suggest a smooth transition between physics across scales. 

Most quantum mechanics problems have classical versions which involve multiple least action solutions. The associated classical multipaths stem either from the initial position or momentum distribution, or from branch points, generated, e.g. by a multiply connected manifold (double slit experiment), by spatial inequality constraints (particle in a box), or by a singularity (Coulomb potential). We show that the exact Schrödinger wave function  can be constructed by combining this classical multi-valued action with the classical density ⁠, computed analytically from  along each extremal action path.

The construction is general and does not involve any semi-classical approximation.

Quantum wave collapse at measurement can be derived from the classical density change. Entanglement corresponds to a sum of classical particle actions mapping to a tensor product of spinors. The results also provide a simpler computational alternative to Feynman path integrals, as they use only a minimal subset of classical paths."

Classical physics can explain quantum weirdness, study shows

On computing quantum waves exactly from classical action (open access)

New study bridges the worlds of classical and quantum physics (original news release) "The weird quantum behavior of subatomic particles can be understood through everyday classical ideas, MIT researchers show."

On computing quantum waves exactly from classical and relativistic action (prepint, first published 5/10/2024, open access)

Math long resisted a digital disruption. AI is poised to change that

Food for thought! I also had the impression that the queen of science was kind of aloof and resisted machine learning & AI to some extent. However, we may see a revolution in the making!

"Mathematician ... is training computers how to prove one of the most famous problems in math history: Fermat’s last theorem.

Resolving the problem isn’t the point. There’s already an accepted proof that was finalized in 1998. That work is a tortuous maze of mathematics that fills about 130 pages over two papers. It spans mathematical fields and unites abstract ideas that previously seemed to have little to say to one another. ... 

Now, the explosion in artificial intelligence has propelled efforts, spearheaded by technology companies, to combine large language models with theorem provers to develop systems capable of autoformalization. In theory, such systems may ultimately be able to do things that humans can’t. ..."

AI could radically change how math proofs are verified "Modern formalization, supercharged by AI, could radically change the way people do mathematics"

The AI Revolution in Math Has Arrived "AI is being used to prove new results at a rapid pace. Mathematicians think this is just the beginning."

AI completes the formal proof of higher-dimensional sphere packing

Good news!

"When Ukrainian mathematician Maryna Viazovska received a Fields Medal—widely regarded as the Nobel Prize for mathematics—in July 2022, it was big news. Not only was she the second woman to accept the honor in the award’s 86-year history ... Today, in a collaboration between humans and AI, Viazovska’s proofs have been formally verified, signaling rapid progress in AI’s abilities to assist with mathematical research. ..."

"These results, originally proved by Maryna Viazovska and collaborators, earned Viazovska the Fields Medal at the 2022 International Congress of Mathematicians. This is the only formalization of a Fields Medal-winning result from this century ..."

AI Proof Verification: Gauss Tackles 24D - IEEE Spectrum

Completing the formal proof of higher-dimensional sphere packing "Using Gauss, we have helped formally verify the sphere packing problem in dimensions 8 and 24 — certifying that the E8 lattice and the Leech lattice achieve the densest possible arrangements of non-overlapping spheres in their respective dimensions."

Fields Medal–winning mathematical work about optimal sphere packing in many dimensions is now formally verified by a human-AI collaboration