On 6 October 2026, OpenAI casually announced it was "releasing a broad range of new mathematical results produced by an internal frontier model", the same system the technology company used to produce its solution to the Navier-Stokes millennium challenge problem a month earlier.

"Broad range" is something of an understatement. The firm's still-internal AI system solved 376 open problems in several fields of mathematics, backed up by autoformalized proofs using the Lean proof assistant which it made publicly available on a GitHub repository.

On 11 May 1959, Eugene Wigner gave a lecture in which he described "the unreasonable effectiveness of mathematics in the natural sciences", pointing out mathematics was uniquely capable of describing observations in the physical world. So much so that abstract mathematical concepts developed without any connection to any physical science were proving capable of describing and predicting physical reality.

While not quite the same, what's happening now with the specially-trained artificial intelligence models making breakthroughs in solving so many open problems at such a torrid pace might be described as the unreasonable effectiveness of AI in mathematics. The rapid development is shaking up the discipline.

A Productivity Explosion

To show how much AI is rattling mathematicians, here's an updated version of a chart we first featured two months ago to illustrate AI's productivity explosion. It shows the number of math papers submitted to the arXiv repository each month from January 1992 through September 2026.

In December 2022, when OpenAI's ChatGPT was first made publicly accessible, arXiv reported some 3,122 math preprint papers had been posted in its database. In July 2026, arXiv reported the posting of 5,696 math papers during the month. For September 2026, arXiv reports 8,919 math papers were added to the archive.

That doesn't include any of the results OpenAI released in October 2026. When we next feature an updated version of this chart, we expect we may have to adjust the vertical scale because the number of newly math papers could exceed 10,000 per month. For its part, arXiv announced it would limit authors to posting no more than two papers per month, but the daily numbers we've seen suggest that is doing little to slow the pace.

Mathematicians Go Into John Henry Mode

In American folklore, John Henry is the story of a former slave who became a "steel driving man" building railroads after the Civil War. In the tale, John Henry took on new steam-powered technology in a contest to find out whether man or machine could do the job of driving steel faster. He won, but died from the exertion.

A team of mathematicians experienced their own version of John Henry's story in September 2026. Quanta Magazine sets the stage for their story:

On the morning of September 11, 2026, Dor Minzer received a text message from a friend asking whether he was close to settling one of the most famous open questions in theoretical computer science. Minzer, a professor at the Massachusetts Institute of Technology and a leading expert on the “unique games” conjecture, initially thought it was a joke. Then more messages started coming in, and the rumors began to cohere. The artificial intelligence company OpenAI, fresh from announcing a bombshell proof about the behavior of fluids that sent ripples through the math world, had allegedly discovered a proof of the unique games conjecture. They might make the result public any day. If Minzer had any results of his own to share, the messages suggested, now would be the time to do so.

The unique games conjecture is an iconic question in computational complexity theory, the study of the inherent difficulty of mathematical problems. Roughly speaking, it states that a problem about satisfying multiple constraints at the same time can be extremely hard, even if you’re willing to settle for a poor approximation of the best possible solution. A proof of the conjecture would also automatically imply that current methods for solving many seemingly unrelated problems can’t be improved. Researchers would be a big step closer to a unified theory of computational difficulty.

Minzer had not, in fact, proved the unique games conjecture. But he and his graduate students Yumou Fei and Shuo Wang had recently proved a milestone result about a closely related question. They were in the midst of writing up their findings — a process that would typically take months or longer — when the rumors began circulating. The trio rushed to finish their draft to avoid being overshadowed by an OpenAI press release. Three days later, prioritizing completeness over clarity, they posted a 95-page paper online with a disclaimer on the first page: “The current version of the manuscript is complete mathematically, but it is not in the shape we wished to share in.”...

“From section 6 onward, there are literally no connecting words,” Minzer said — just an unbroken sequence of definitions and proofs of intermediate results. “We felt the need to apologize for that.”

It's not as tragic an ending as John Henry's story, yet it is telling all the same. For mathematicians who until recently haven't had any need for urgency in doing what they do, AI is pressing them to become more productive than they've ever been before.

How will they adapt? And for that matter, how many other professions will soon find themselves in a similar situation?

References

arXiv. Math Submissions. [Online Database]. Accessed 15 October 2026.

arXiv. Monthly Submissions. [Online Article and CSV Data]. Accessed 15 October 2026.

ScriptByAI. OpenAI & ChatGPT Timeline: GPT Release Dates to GPT-5.6 (2026). [Online Database]. Accessed 20 August 2026.

Image Credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a dumptruck labeled 'OpenAI' depositing hundreds of math papers in front of mathematicians who look on sadly and worry about their future".

In cracking one of the biggest unresolved problems in mathematics, OpenAI launched a controversy over the information it used to train the developmental artificial intelligence system behind it, which raises questions over to whom credit for the accomplishment belongs. The controversy is overshadowing AI's more important contribution to the advancement of mathematics: its role in "autoformalizing" mathematical proofs in the Lean proof assistant.

Autoformalization refers to the automated formal verification of a mathematical proof utilizing proof assistant software. That matters because developing the computer code to verify a proof is often a time consuming and often tedious task for mathematicians.

As a general rule of thumb, it takes 40 hours of labor for a mathematician to formalize the equivalent of one page of an established mathematical proof presented in a textbook with all supporting material preceding it using the popular Lean proof assistant. John D. Cook, an applied mathematician and statistician who runs a consulting firm, estimates a research paper in mathematics takes about 20 times more effort to formalize into a Lean-verified proof, mainly because time needed to collect and validate all the supporting material for it, which may or may not be presented within the paper.

OpenAI's Navier-Stokes proof runs 166 pages. By Cook's back-of-the-envelope math, formalizing all the math needed to verify it could take as much as 132,800 hours for mathematicians to execute.

OpenAI autoformalized its Navier-Stokes proof in just 17 hours.

It's also not just OpenAI's technology making these kinds of advancements toward automating the most tedious and time consuming aspect of modern mathematics. On 4 September 2026, less than a week before OpenAI unveiled its Navier-Stokes proof, Anthropic announced its Claude AI system had successfully generated Lean-verified code for Andrew Wiles' proof of Fermat's Last Theorem (FLT). In doing so, Anthropic beat the mathematicians who had been working for years to formalize it by manually coding it in Lean.

Here's how Anthropic describes what its Claude AI system accomplished:

One way to check a proof’s correctness is to ask a computer to do it. Proof assistants like Lean verify the logic of a proof algorithmically, demonstrating its correctness beyond a doubt. The difficult part for humans is rewriting the proof so Lean can understand it. While a proof written for human readers will skip many obvious steps, Lean needs to see every step, no matter how trivial. Human proofs also build on centuries of published work, while a formalization starts from the tiny fraction of math that’s been formalized already.

For FLT, the formalization process was expected to take years. Just the blueprint the mathematical community has been using to describe the initial phase of the project runs to 86 pages.

Claude completed the proof in 11 days, producing computer-verifiable proofs of 30,300 theorems along the way (using 29,500 in the final proof). Dozens of Claude agents collaborated to define concepts, prove intermediate theorems, and use those theorems to prove ever harder statements. At 13 million lines of Lean code, Claude’s proof is over 5x the size of Mathlib, the principal community library of mathematical proofs this theorem builds on.

Here's some more back of the envelope math. If mathematicians were given $20 million and told to generate the Lean code to verify the Navier-Stokes proof using 132,800 hours of labor, no more and no less, assuming they got it done, they would effectively be paid a little over $150 for each hour of labor. That may even be close to what it would actually cost a single trained mathematician or coder per hour of their labor. Although assuming they worked 40 hours a week, 50 weeks a year, it would also take them nearly 64 years to perform the task.

How many trained mathematicians do you think it might take to accomplish the same task in 17 hours? Keep in mind that level of execution would also take extraordinary planning, preparation, coordination, and execution on their part to accomplish the task in that time. Do you think that extraordinary effort would cost more or less than $20 million?

Businesses around the world are already running numbers like these. The potential to realize massive savings in time and cost is why AI technology has sparked an investing and development boom in the last few years. It might have cost OpenAI $20 million to develop its AI systems to crack the Navier-Stokes equations including generating the Lean code to verify the proof, which perhaps appears excessive next to the Clay Mathematical Institute's $1 million prize for the achievement. But how much would the alternative of having an army of trained mathematicians to do the same job in the same time have cost?

Previously on Political Calculations

We've been covering developments toward the resolving the open question of when the Navier-Stokes equations describing fluid motion works and when it doesn't for some time. Here's our coverage in chronological order:

• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• Navier-Stokes and the Emergence of Order from Disorder
• What is Emergence?
• The Biggest Math Story of 2019
• A Proof for Batchelor's Law
• The Biggest Math Story of 2022
• Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales
• The Biggest Math Story of 2025
• AI Cracks the Navier-Stokes Millennium Problem, Part 1
• AI Cracks the Navier-Stokes Millennium Problem, Part 2

On 8 September 2026, OpenAI officially announced it had determined the Navier-Stokes differential equations for describing the motion of fluids can develop a "singularity", or "blow up" or "break" to use more expressive terms, while running in a finite period of time.

Here's how they described the open question about the math involved in the Navier-Stokes equations:

The Navier–Stokes equations use Newton’s second law of motion (“F=ma”) to describe how fluids move. Importantly, they treat a fluid as a continuous medium rather than tracking individual molecules. These equations are used for aircraft design, weather forecasting, and the study of blood flow.

A fundamental open question for these dynamical equations has been whether the continuum approximation of the fluid can break down. Specifically, can the Navier–Stokes equations for a three-dimensional incompressible fluid with constant density develop a “singularity,” even when the motion starts smoothly? Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time. The development of a singularity would have to happen despite the presence of viscosity, which tends to smooth out motion. Because a real fluid cannot move infinitely fast, this would mark a breakdown in how the equations model the fluid. To continue modeling the system, one would then need to track the behaviour of each particle individually.

The equations date to the nineteenth-century work of Claude-Louis Navier and George Gabriel Stokes. In 1934, Jean Leray proved that solutions exist in a generalized sense, but whether they always remain smooth became a central unanswered question. In 2000, the Clay Mathematics Institute named the Navier–Stokes existence and smoothness problem one of seven Millennium Prize Problems.

And here's their result:

Our system produced an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time. The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity. This resolves the Navier–Stokes Millennium Prize problem by establishing statement “C” (and also “D”) in the official Millennium Prize formulation⁠.

The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti. This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics. The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand. More mathematically, the terms in the Navier–Stokes equations that describe the motion—acceleration, pressure gradients, momentum transfer, viscosity—must both become big yet cancel in a precise way. This detailed balance leaves a smooth external force even as the velocity of the fluid grows without bound.

Assuming it holds, OpenAI's solution to the Navier-Stokes Millennium Prize challenge, for which it might win $1 million, will have cost the firm $20 million.

The Clay Mathematics Institute, which established the "Millennium Prize" challenges for solving seven long-standing open problems in mathematics back in 2000, issued a statement indicating the million dollar prize for cracking the Navier-Stokes millennium problem is now pending their official verification.

Controversy soon erupted because of questions about whose mathematical research OpenAI's still-internal AI system used to develop their proof the Navier-Stokes equations will not always work and that they can break down under specific circumstances. The following Code Report from Fireship provides an entertaining rundown of the conflict:

Terry Tao's blog post announcing the accomplishment and assigning credit as best as could be done on 7 September 2026 is here.

The controversy over credit has the potential to arrest the rapid progress in mathematics that is being boosted by AI technologies. That potential exists because of the AI systems' ability to rapidly absorb new developments by mathematicians and apply them to a multitude of other problems without really understanding the work. In doing so, the AI developers are creating a strong, adverse incentive for the mathematicians to hold back their work until it has been fully verified to ensure they retain credit for it.

On 11 September 2026, Tao signed onto a declaration with other Fields Medal-winners (the math equivalent of the Nobel prize for science disciplines) who are alarmed at the major disconnect they're seeing between mathematicians and AI developers, which they argue puts advances in mathematics at risk.

That's a shame because the controversy is obscuring one of the biggest math stories of the year, which points to the massive capabilties that AI technologies are unlocking to advance mathematics. We'll cover that part of the story in Part 2.

References

Tristan Buckmaster. Announcement of three results with Levent Alpöge on finite-time blowup with smooth forcing for incompressible porous media, for Boussinesq, and for 3d incompressible Euler. [Mastodon post]. 7 September 2026.

Terence Tao. Finite time blowup with smooth forcing term for the incompressible porous medium, Boussinesq, and incompressible Euler equations. [Online article]. 7 September 2026.

OpenAI. On the Navier-Stokes Millennium Prize Problem. [Online article]. 8 September 2026.

Complexity. Visualizing the OpenAI solution to the Navier Stokes Equation. [Online video]. 10 September 2026.

Clay Mathematics Institute. Navier-Stokes Announcement. [Online article]. 11 September 2026.

Math and AI. A Severe Misalignment of AI in Mathematics. [Online article]. 11 September 2026.

Previously on Political Calculations

We've been covering developments toward resolving the open question of when the Navier-Stokes equations describing fluid motion will work and when it won't for some time. Here's our coverage in chronological order, the articles marked with an asterisk are directly relevant to the resolution of the Clay Mathematical Institute's Navier-Stokes Millennium Problem:

• The Biggest Math Story of 2017*
• Navier-Stokes and the Emergence of Order from Disorder
• The Biggest Math Story of 2019
• A Proof for Batchelor's Law
• The Biggest Math Story of 2022*
• Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales
• The Biggest Math Story of 2025
• AI Cracks the Navier-Stokes Millennium Problem, Part 1*

We recently discussed how advances in technology are contributing to boosting the productivity of mathematicians during the last thirty years. But the chart we featured in that article depicting how the number of math papers being published each month has nearly doubled in the last three years may not do full justice to how fast the pace of new papers coming out has changed.

Another way to communicate that acceleration in output is Matt Parker's approach of focusing on a small subset of the new papers being published in the last few months to show how they represent a big increase in the capability of the AI large language models used to generate their findings. In the following video, in what is becoming an annual tradition, he interrupts his vacation to cover breaking news in the world of mathematics.

Parker makes an explicit point of noting the use of the Lean proof assistant to verify their findings, which gives greater confidence the findings in these papers will stand up to serious scrutiny.

Not all the preprint papers that have recently populated the mathematics category of the arXiv database meet that standard, as Reddit math community poster Salt_Attorney recently observed.

By contrast, Lean verification was used in the ten advances spanning several disciplines in mathematics OpenAI claims its Astra AI system has made. As such, they have a much better chance they'll be found valid when the findings are reviewed.

But the question now being raised is whether verification of findings by a proof assistant is enough to accept AI-generated results as presented. For further reading on that topic, we'll recommend Gary Marcus' discussion of OpenAI's claimed accomplishment to appreciate it may not be either as grand or the Astra AI system as capable as presented in OpenAI's announcement.

It may not seem like it, but the millennia-old academic discipline of mathematics has been undergoing a productivity revolution during the last thirty years.

That revolution is being enabled by the adoption of several new technologies, each of which is following a S-shaped logistic growth curve model. Mike Roberts of Strategic Tool Kits describes the pattern of how technological advancements increase performance over time:

The S curve is a strategic concept that describes how the old ways mature and are superseded by new ways.

In the early days of new technology, it takes a long time to improve performance. People are working out the technology, and the applications, ironing out the flaws, and building the ecosystem. Slowly over time, performance accelerates.

After a certain time, the rate of improvement hits a peak and then starts to slow down. Easy wins have all been made and the learning curve has been fully ridden for continuous improvement. Some of the fundamental barriers of the technology are reached. Eventually, the improvement tapers off and a plateau with this technology is reached. This is the “S” shaped curve.

When a new productivity-enhancing technology comes along after the older technology reaches its mature phase, the process repeats with performance building on its old plateau and rising to new heights.

For mathematicians, the two great technological improvements of the twenty-first century have been the widespread adoption of the arXiv database for publishing preprint papers to more quickly communicate their discoveries and the more recent development of Artificial Intelligence (AI) systems paired with proof assistant systems like Lean that are helping automate large portions of their work.

ArXiv provides data on the number of new papers that are uploaded to its preprint paper database each month. The following chart reveals its data for math papers, not counting cross listings from other categories of academic papers, from January 1992 through July 2026:

The data shows adoption of the arXiv database for communicating advancements in maths went through the full S-curve advancement cycle, reaching a fully mature phase by the early 2020s with output holding fairly level for several years. But after ChatGPT (3.5) was launched at the end of November 2022, the Large Language Model (LLM) technology facilitiated a steady increase in the productive output of mathematicians for the first time in years. By making it possible to automate a portion of writing academic papers, the number of papers published to arXiv each month began to rise in a steady, linear trend.

But soon after OpenAI's o3-mini LLM-based reasoning model was released on 31 January 2025, the rate of output for publishing new papers to arXiv exploded in what appears to be the exponential growth phase of the S-shaped logistic growth pattern. The new cost efficient o3-mini reasoning model was specifically developed to automate analytical tasks in Science, Technology, Engineering, and Mathematics fields and also coding.

For mathematics, the new AI technologies make rapid advancements possible by drawing on the body of published work, like the thousands of math preprint papers documented in the arXiv database and other resources that catalogued large numbers of unresolved conjectures in digital-friendly formats, to test new possible ways of testing those conjectures. When promising proofs or disproofs of the conjectures are identified, the technology automates their verification using the proof assistant software technology that has also come into its own during this period. Using proof assistants also has the benefit of coding any new successful proofs into their proof libraries, which can then be mixed and matched as needed to test other conjectures.

When that's done, the remaining step for the mathematicians orchestrating what's effectively become a massive, automated collaboration exercise is to write up a new preprint paper and publish it to the arXiv database.

Mathematics isn't the only field experiencing a boom in preprint papers. Economics has likewise seen the number of papers published each month almost double since ChatGPT 3.5's public release. Unlike math however, economists don't have the equivalent of maths' proof assistants to verify their findings, which raises questions about the quality and validity of the flood of new papers in the field.

References

arXiv. Math Submissions. [Online Database]. Accessed 8 August 2026.

arXiv. Monthly Submissions. [Online Article and CSV Data]. Accessed 8 August 2026.

Ecology.net. Logistic Growth. [Online Article]. 15 December 2025.

Pablo Groisman. Math papers uploaded to arXiv per month, January 1992 to July 2026. [Online Article]. 3 August 2026. [Our article was inspired by Pablo's chart!]

Mike Roberts. S Curve – What is it? [Online Article]. 13 July 2024.

ScriptByAI. OpenAI & ChatGPT Timeline: GPT Release Dates to GPT-5.6 (2026). [Online Database]. Accessed 20 August 2026.

Image credit: S Curve Concept in Science by Oliver Tacke on Wikiversity. Creative Commons CC BY-SA 3.0 Attribution-ShareAlike 3.0 Unported Deed. 17 July 2012.

Knots hold a special place in the hearts of sailors and mathematicians.

That's almost self-explanatory for sailors, who have been tying knots as an essential part of their craft for centuries. They even used them to tell how fast they're going over open water.

Mathematicians' fascination with knots is harder to explain, but comes down to their love of counting. The more complicated the knot, the more the knots strands cross each other, the better. And what better way to make a knot more complicated than by taking two knots, cutting each and then joining their open ends together to make an even bigger, more complicated knot?

They even defined a special rule about the practice to calculate how much more complicated the resulting knot would be. They conjectured that if they took the unknotting number for each knot, which is to say the number of steps it would take to tranform the knot into a simple loop (imaginatively called the "unknot"), and added them together, the result would be the unknotting number of the combined, more complicated knot.

This additivity conjecture worked with just about every combination of knots they could throw at it. Until two mathematicians discovered an example where that rule didn't work. Instead of leading to a more complicated knot, they found a knot that became easier to unknot after being joined. It didn't add up as previous generations of mathematicians had believed it should, and because it didn't, they proved the conjecture about the additivity of unknotting numbers is untrue for all cases.

In the following video, Trefor Bazett guides viewers through basic knot theory in an easy to understand presentation before getting to the remarkable disproof of the additivity conjecture in knot theory by Mark Brittenham and Susan Hermiller, which only took them ten years to work out:

Brittenham and Hermiller's paper is here. Bazett has a second video featuring an extended interview with Brittenham and Hermiller.

The knot they found that broke the rule is the (2,7) Torus Knot, an example of which is the featured illustration for this article. It's also known as the 7₁ knot in a different mathematical knot nomenclature system, but we're not going to get into that topic because why complicate knots any more than needed?

Image credit: Blue 7₁ Knot or (2,7) Torus Knot by Jim.belk on Wikimedia Commons. Public Domain.

In the last decade, proof assistants have revolutionized how mathematicians establish whether a mathematical theory is valid. One in particular, called Lean, has risen to the forefront of the field. Its productivity-enhancing capabilities are behind several notable proofs that have been demonstrated in recent years. What's more, its integration with Artificial Intelligence technologies is contributing to a rapid pace of new advances in solving long standing but, until this year, unproven mathematical conjectures.

Much of that story is now being told in Kevin Hartnett's The Proof in the Code: How a Truth Machine Is Transforming Math and AI, which is proving to be an exceptionally well-timed book. It captures the short history of the rapid development of the Lean proof assistant, which is contributing to a revolution in how mathematicians do what they do. Hartnett's book is all the more remarkable because it doesn't require readers to have an extensive background in either mathematics or computer science to both follow the story and understand its significance.

And what a significant story it is. Here's a short summary of how Lean is aiding the advancement of mathematics:

• It can automate large parts of the process of verifying mathematical proofs.
• Its results are trustworthy. If a proof is validated in Lean, the proof will hold.
• Its library allows mathematicians to take advantage of proofs developed by their peers, using them as modular building blocks to help establish new proofs.
• It can handle very sophisticated proofs developed by the world's top mathematicians.
• It facilitates massive collaboration efforts among mathematicians and other contributors.
• It can be integrated with Artificial Intelligence (AI) technologies, such as reinforced learning techniques and Large Language Models, which can both improve its performance and the performance of the AI systems.

All of these things underscore the role of Lean as an amazing productivity enhancing software tool for its users. But it wasn't always that way. Perhaps even more remarkably, Lean didn't start out as a tool aimed at helping mathematicians.

As originally envisioned by Leo De Moura, who wrote several generations of its code, it was supposed to be a tool to help software developers find and fix bugs before they released software. A chance discussion with mathematician Jeremy Avigad in 2013 identified the potential of the code to aid mathematicians in developing their proofs, which ultimately set the direction of Lean in motion.

The rest of Harnett's story is how their efforts pulled in other key players who recognized its potential and worked through its development challenges to make it useful first, then to make it more and more capable.

For a story that involves many mathematicians and the math they were seeking to validate, Harnett keeps mathematical equations and symbols to a minimum. That's a vital requirement in making the story accessible to a general audience and showcases Hartnett's experience as a math writer. It's not until the later chapters of the book that mathematical statements begin appearing in the text, which are backed by plain language descriptions, which makes them very approachable.

Even more remarkably, the only example of Lean code that Hartnett presents is contained in the book's Appendix. The example is Lean's version of a 2,300+ year-old proof by Euclid that confirms there is always another larger prime number, which is to say that prime numbers extend into infinity.

If you only read one math book this year, this is the one to read. It's a fantastic introduction to some of the most advanced happenings going on in the world of maths and computer science and what is becoming possible today that hasn't been before. Highly recommended!

If you're interested in more discussion, Hartnett was recently interviewed by Breaking Math's Autumn Phaneuf and Noah Giansiracusa. Here's the video of the interview:

If you look at the back of a box of Kellogg's Frosted Flakes Glazed Donut Holes cereal, you'll find a startling claim:

Matt Parker investigated and came to three remarkable conclusions:

1. Kellogg's has a typo in their formula for the surface area of a donut.
2. He is the kind of guy who will pay someone to record of a video of himself eating a bowl of cereal."
3. Despite the typo, Kellogg's is right!

All this and more is revealed in the following video of Parker's investigation:

Kellogg's claim would not be true if the thickness of the glaze layer was the same between the spherical donut hole and toroidal donut geometries. But the reason Kellogg's claim is able to hold is because they really loaded up the amount of glaze onto their cereal-based donut hole structure, making its coating much thicker than what can be achieved by applying the same mass of sugary glaze spread out over the larger surface area of a donut (or in math terms, a "torus"). In effect, they maxxed out the ratio of glaze to cereal.

But the more amazing thing is that Kellogg's has not fixed the typo in the formula for the surface area of a donut-shaped object, even though is has been printed on every box of its Glazed Donut Holes cereals for more than a year after it was first reported!

There are quite a few famous unsolved problems in mathematics. Some are so famous they have million dollar bounties on them that will be paid out to the first mathematicians who definitively either prove or disprove them.

And then there are other important conjectures out there that, in addition to academic recognition, might pay smaller rewards to the people who crack them. But which if they are cracked, can be even more valuable if they can successfully be put to use.

One of those smaller conjectures is the Talagrand Convexity Conjecture. Proposed by French mathematician Michael Talagrand in 1995, Talagrand famously offered a $2,000 prize to anyone who solved it. Here is Jules Aknin's simplified description of the conjecture:

Talagrand’s convexity conjecture is a statement about high‑dimensional geometry: even in enormous, messy clouds of points, simple convex shapes are guaranteed to appear. In other words, no matter how chaotic the configuration looks, there is unavoidable structure hiding inside it.

If such structures are truly unavoidable, they can open the door to practical applications, including in financial markets. Aknin describes how they might contribute to finding order within chaos:

At a deeper level, results like this sit exactly at the intersection of geometry and probability. They don’t just create “pretty shapes” in higher dimensions, they show how structure emerges inside high‑dimensional random systems, which is the same conceptual problem we face when building models for complex datasets. In fact, as the original article notes, this kind of unification between geometric and probabilistic thinking could eventually influence how machines process high‑dimensional data and how we design algorithms that operate in those spaces.

[...]

On the surface, a high‑dimensional cloud of points in geometry and the Australian equity market do not look related. But in practice, we deal with a very similar object every day: hundreds of stocks, multiple time horizons, momentum profiles, risk factors, macro shocks and behavioural flows all interacting at once.

From a distance, that system may appear as pure noise. Talagrand’s result reinforces an idea that systematic managers have believed for a long time: structure is not optional, it is inevitable; the real question is whether you have the tools and discipline to find it.

Even though it doesn't carry the million-dollar prize of mathematics most famous unsolved problems, the stakes for those who can successfully build on the proven conjecture are still very high. That work is quite possibly worth a lot more than Talagrand's two-thousand dollar prize for the proof itself.

A team of three mathematicians stand to collect Talagrand's $2,000. Merrick Hua, Antoine Song, and Stefan Tudose posted a preprint paper of their proof on 11 May 2026, in which they converted the problem from one involving geometry to one involving probability and combinatorics. Like many math stories this year, AI is involved, but unlike those other stories, it's only a bit player.

The new proof was worked out by Dongming Hua and Antoine Song from the California Institute of Technology, and Stefan Tudose from Princeton University, who joined the other authors after hearing about their work. Together, the mathematicians reformulated Talagrand's geometric conjecture to a problem of probability theory and random vectors. In their paper published on the arXiv preprint server, they proved an equivalent conjecture for probability, showing that any 1-subgaussian random vector in n dimensions can be expressed as the sum of three standard Gaussian random vectors.

This result solves Talagrand's convexity problem, proving that for any large enough set in Gaussian space, a convex set of significant measures can be found inside a triple sum of the original set. The solution also confirms a combinatorial analog of the problem, which is important for discrete mathematics.

Initially, Song and Hua say they attempted to work out a solution with the help of ChatGPT. However, while the LLM helped to answer some of their questions and move them closer to a solution, it was Tudose who provided the final proof. Ultimately, the team did not use the work done with ChatGPT. In their paper, the team writes that Tudose's proof was "more general and conceptual."

It's a remarkable achievement. It is also a harbinger of the kind of role AI will find as just another tool used by people.

Image credit: Texture of an iced leaf image by Daniela deGol on Wikimedia Commons. Creative Commons CC By 4.0 Attribution 4.0 International Deed.

Prime numbers are unique. Unlike all other numbers, prime numbers cannot be divided equally by a whole number to get a whole number result except for itself and the number 1.

By contrast, every other number but prime numbers will include other numbers as factors. Also called composite numbers, when you drill down far enough, you'll find each has a unique factorization made up of nothing but prime numbers.

If you start counting up from 1, you'll frequently run into prime numbers at the beginning. But as you count higher and higher, you'll find prime numbers become fewer and farther apart. At first glance, it seems like they're randomly distributed among all the numbers you're counting. But that appearance is deceptive, because when you tease the numbers just right, a non-random pattern emerges.

The following video by Physics Explained's Rhett Allain is one of the best we've seen in establishing the foundation of just how the numbers have to be teased to reveal the pattern that the prime numbers are following.

In the video, New Scientist's Jacklin Kwan focuses on the Riemann Hypothesis, which describes the pattern that prime numbers appear to follow into infinity, which is the biggest unproven conjecture in math:

Previously on Political Calculations

• An Introduction to the Riemann Hypothesis

Increasingly capable Artificial Intelligence (AI) technologies are gaining steam in proving long-standing mathematical conjectures. The latest development involves a proof of another one of prolific mathematician Paul Erdős' 1,135 unsolved problems, but unlike AI's previous accomplishments in tackling part of Erdős' legacy, GPT-5.4's proof of Erdős #1196 appears to be genuinely novel.

The previous proofs by AI systems of other Erdős problems did not answer the question of whether the technology was bringing anything new to the table. The problems themselves could be considered "low-hanging fruit", whose unsolved status had more to do with the obscurity of the conjectures within the Erdős collection than their difficulty. When AI developed proofs for them, it more or less followed the playbook that mathematicians had established in proving other Erdős conjectures.

But Erdős' 1196th conjecture is not in that category. Mathematicians had previously taken on the challenge of developing a proof for it, largely turning to the tools of probability and statistics in the process. But instead of copying that approach, GPT-5.4 Pro found the path for proving the conjecture differently. Mathematician Terrence Tao offered this observation:

I had previously stated the opinion that the AI-generated proof had inadvertently highlighted a tighter connection between the anatomy of integers and the theory of Markov chains than had previously been explicitly noted in the literature. Based on further developments, I would like to update that opinion to the following: the AI-generated proof artefact, when combined with subsequent (and mostly human-generated) analysis, has revealed a tight connection between the anatomy of integers and flow network theory that does not, to my knowledge, have any explicit precursor in the literature (although related uses of Markov chains in adjacent settings do appear in that literature).

The development of the proof was verified using the Lean proof assistant, which we would argue is the secret sauce behind why AI technologies are making such rapid progress in advancing proofs to unsolved problems in the field. The pairing of the technologies is key to the advancement.

Getting back to the novelty of the proof, here's a comment by Jared Duker Lichtman, who developed the first proof of the related Erdős primitive set conjecture as part of his doctoral thesis in 2024.

In my doctorate, I proved the Erdős Primitive Set Conjecture, showing that the primes themselves are maximal among all primitive sets.

This problem will always be in my heart: I worked on it for 4 years (even when my mentors recommended against it!) and loved every minute of it.

[Primitive sets are a vast generalization of the prime numbers: A set S is called primitive if no number in S divides another.]

Now Erdős#1196 is an asymptotic version of Erdős' conjecture, for primitive sets of "large" numbers. It was posed in 1966 by the Hungarian legends Paul Erdős, András Sárközy, and Endre Szemerédi.

I'd been working on it for many years, and consulted/badgered many experts about it, including my mentors Carl Pomerance and James Maynard.

The proof produced by GPT5.4 Pro was quite surprising, since it rejected the "gambit" that was implicit in all works on the subject since Erdős' original 1935 paper. The idea to pass from analysis to probability was so natural & tempting from a human-conceptual point of view, that it obscured a technical possibility to retain (efficient, yet counter-intuitve) analytic terminology throughout, by use of the von Mangoldt function \Lambda(n).

The closest analogy I would give would be that the main openings in chess were well-studied, but AI discovers a new opening line that had been overlooked based on human aesthetics and convention.

In fact, the von Mangoldt function itself is celebrated for it's connection to primes and the Riemann zeta function--but its piecewise definition appears to be odd and unmotivated to students seeing it for the first time. By the same token, in Erdős#1196, the von Mangoldt weights seem odd and unmotivated but turn out to cleverly encode a fundamental identity \sum_{q|n}\Lambda(q) = \log n, which is equivalent to unique factorization of n into primes. This is the exact trick that breaks the analytic issues arising in the "usual opening".

Joshua Zelinsky offers perhaps the best framing of the accomplishment and what it could mean for additional progress:

Four things to note: #1196 is a decently well known problem. It wasn’t like Erdős-Straus level fame, but it is well known enough that I was familiar with it. Second, this is not a problem where no one had worked on it; there was a lot of prior work on it and closely related problems. Third, this is not example where the AI made small modifications to things in the literature or recognized that large parts of the problem were in an obscure paper. The approach the AI used is largely a different direction than the literature on this problem went. Fourth, and closely related to three, this proof does look like parts of it will inspire subsequent proofs because it really is going in a different direction which now looks likely to be a productive line of investigation for similar problems.

If this kind of progress continues, the productivity of AI technology in cracking unsolved math problems will be the biggest math story of the year. AI was already the biggest math story of 2025, but 2026 is shaping up to be even more so.

Once upon a time, and in truth, as recently as four years ago, there was a strong relationship between the spot price of gold and the inflation-indexed market yield of 10-Year Constant Maturity U.S. Treasuries.

It was an inverse relationship. When the inflation-adjusted interest rate on the 10-year bonds fell, signaling an increase in inflation, the price of gold would rise. And vice-versa. If the inflation-adjusted yields of these treasuries rose, indicating falling inflation, the price of gold would fall as well.

That made a sort of sense. But that relationship has broken down in the last four years. Starting from 17 March 2022, when the Federal Reserve finally acted to hike interest rates to combat the high inflation unleashed by the Biden administration a year earlier, the relationship between the inflation-indexed 10-year Treasury and gold spot prices has steadily broken down.

We can see that in the following chart, in which the price of gold has fully decouples from the interest rate of the inflation-protected 10-year Treasury.

Most of this decoupling has taken place since 27 December 2023. At that time, the yield of the inflation-indexed 10-year Treasury was 1.64% and the price of an ounce of gold was $2,079. Since that date, the 10-year inflation-protected Treasury has ranged between that low and a high of 2.28% on 30 April 2024. Today, that yield is just below the middle of that range at 1.88%.

But the price of gold has soared during this time. In recent months, as the 10-year TIPS yield has ranged between, it soared to reach a high of $5,414.49 an ounce on 28 January 2026. In the weeks since, it has plummeted, losing over 10% of its peak value. All without any big change in the inflation-protected treasury yield.

It's like the carnival game "high striker". Gold prices are rising and falling by huge amounts independently of changes in yields and inflation.

In mathematics, the slope of a vertical line is described as "undefined". Which is to say there is no relationship between it and whatever the horizontal axis represents. In the case of gold prices, inflation, and the yield of 10-year U.S. Treasuries, we can say that in March 2026, no such relationship exists between these three things, nor has there been such a relationship in years.

Perhaps there never was. Or perhaps we're missing a bigger factor that's holding greater sway today. If we are, what do you suppose it is and why has it been able to cause the price of gold to change so much in during the last four years as compared to how it changed during the preceding 15 years?

Image Credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon of a Wall Street bull and bear who are taking turns playing the high striker carnival game that is labeled 'GOLD PRICES' and the bear asks 'DIDN'T THIS GAME USE TO HAVE SOMETHING TO DO WITH INFLATION?".

Recent months have seen several announcements in which the Large Language Models (LLM) behind modern Artificial Intelligence (AI) technologies have solved long-standing problems in the field. Among those previously unsolved problems are a handful originally conceived by Paul Erdős, who generated a catalog of 1,135 challenges for mathematicians to take on. Several AI developers target these problems as means of measuring the progress they're making in developing their systems.

But are today's AI systems really capable of solving these unsolved problems? LLMs have been described as an advanced form of an autocomplete function, or even as a "glorified autocorrect" program the AI systems use to statistically predict what response should follow the prompts it has been given based on the reams of data on which it has been trained.

In the case of the Erdős problems that have been solved by AI, there could be something to that argument. The solved problems have been described as being relatively low-hanging fruit, whose unsolved status may have more to with their relative obscurity. Being similar to other Erdős problems that have been solved, which would be part of the training library used by the LLMs, that similarity could be enough to solve them.

The alternative hypothesis is the math-LLMs are genuinely capable of coming up with original solutions for these problems. But how can we tell which hypothesis is closer to the truth?

A group of eleven mathematicians has proposed a interesting experiment to find out. They're tapping their currently unpublished research to remove the possibility that the LLM is effectively rehashing mathematical solution processes to which they have previously been exposed. Here's the abstract for their preprint paper, which they uploaded on 6 February 2026:

To assess the ability of current AI systems to correctly answer research-level mathematics questions, we share a set of ten math questions which have arisen naturally in the research process of the authors. The questions had not been shared publicly until now; the answers are known to the authors of the questions but will remain encrypted for a short time.

"Short time" was one week. The ten questions were unencrypted on 13 February 2026.

That action started a clock for challenging today's math-AI systems to see if they're genuinely capable of autonomously solving mathematical research questions. At this writing, we don't know what results, if any, have been put forward and whether they stands up to scrutiny.

Regardless of how it goes, it's a genuinely exciting research effort for which we're looking forward to learning the outcome.

Image credit: Artificial Intelligence - hand touches icon with symbol of network and brain by Gerd Altmann on PublicDomainPictures.com. Creative Commons Creative Commons - CC0 Public Domain.

Shiraz Robinson is a grad student in data science at the University of Virginia, who came to our attention shortly after the Christmas holiday for his Merry Christmans equations. Unfortunately, that was too late to feature his tradition of blending math formulas and holiday celebrations together, at least for that holiday, but he's back in time for Valentine's Day with another holiday-appropriate math formula!

Here's the equation, suitable for sharing with that special someone who appreciates math as much as you do! And before you ask, yes, it's safe for work....

x² + [y - (x²)¹^/³]² = 1

While the mathematically inclined will immediately interpret this equation of love, others may need to plot it on a graph to truly appreciate its romantic message. We recommend checking out this GeoGebra version because you can play with the parameters d'amour.

Update 18 February 2026: Or, if you prefer to leave the math to others (and use a fancier equation), there's this video!

Previously on Political Calculations

• Picking the Right Date for Valentine's Day
• The Indisputable Inter-Age Rule for Dating
• Happy Valentine's Day!
• The Maths of Love
• The Most Romantic Shape in Mathematics
• A Formula for Love
• Conway's Game of Love

Mathematician Paul Erdős died nearly 30 years ago, but left behind a massive legacy. Famously known for his collaborations with other mathematicians and his eccentric practice of showing up unannounced to their doorsteps and staying only as long as they provided him with interesting challenges to ponder, the prolific mathematician left behind 1,135 problems for succeeding mathematicians to advance the field by either proving or disproving them.

In the last few months, three of those problems have been cracked by Artificial Intelligence (AI) systems harnessed with the well established Lean proof assistant. NeuronDaily tells the story of the third, which is impressive because there was little-to-no online material the AI system could have accessed to develop its proof, which means the AI paired with a proof assistant is generating genuinely original work:

Remember those brain-teaser math problems from school that made you want to throw your pencil? Now imagine ones so hard they've stumped mathematicians for decades. Paul Erdős, who published more papers than anyone in math history, left behind hundreds of these puzzles when he died in 1996. This weekend, GPT-5.2 Pro solved one.

Neel Somani prompted the AI to tackle Erdős Problem #397, which asks whether infinitely many solutions exist for a specific equation involving central binomial coefficients. GPT-5.2 generated the proof, the tool Aristotle formalized it in Lean (a verification language), and Fields Medalist Terence Tao accepted it.

Here's Neel Somani's X tweet announcing the accomplishment:

NeuronDaily explains why the proof of Erdős Problem #397 is a big deal:

It’s part of a wave of autonomous solves: GPT-5.2 has now cracked Problem #728, #729, and 397.

Verified mathematics: The Aristotle system auto-corrected gaps in proofs and produced Lean-verified code.

Self-contained reasoning: Unlike October 2025's GPT-5 controversy (which just found existing literature), Tao says these are original proofs.

While impressive for the technology, the new proofs don't yet represent the kind of ground-breaking accomplishment that would permanently leave human mathematicians in the dust.

The catch? Tao emphasizes these are “lowest hanging fruit”; problems solvable with standard techniques, not profound breakthroughs. GPT-5.2 scores 77% on competition-level math but only 25% on open-ended research requiring genuine insight.

Three years ago, having any computing system score 77% on competition-level math would have been a remarkable achievement, but scoring 25% on open-ended research was a pipe dream. Considering the amazing progress that's been made with the use of AI in maths in just the past three years, what lies in store for the next three years?

Image credit: Paul Erdős and Terrence Tao in 1985. Wikimedia Commons. Creative Commons Attribution-Share Alike 2.0 Generic Deed. We were surprised that Erdős collaborated with Tao when the latter was just 10 years old! Terence Tao, of course, went on to have an equally remarkable career and has an Erdős Number of 2.

The year that was 2025 has all but come and gone, so it's the perfect time to look back at the year's biggest math stories!

For our selection criteria, we've emphasized math stories that involve practical applications in selecting both the contenders and the Biggest Math Story of 2025, which we'll present at the end of this year's edition. That real world connection is something we find essential in exploring achievements in math, which is why we place our main focus on it.

But that doesn't stop us from appreciating the year's "pure" math accomplishments. For example, we're fans of the work by the volunteers behind the Great Internet Mersenne Prime Search project, which is dedicated to identifying these special prime numbers. Last year, they made news when a volunteer found the current record holder for the largest known prime number, M(136,279,841), the 52nd Mersenne prime.

While they didn't find a bigger prime number this year, their work to verify that no other Mersenne primes fit in-between the gaps of their previous discoveries continued. In 2025, they expanded their test results to verify that they had identified all Mersenne primes smaller than M(77,232,917), the 50th Mersenne prime. We anticipate they'll clear the distance between that prime and M(82,589,933), the 51st Mersenne prime, sometime in 2026.

There is however an enormous gap between the 51st Mersenne prime and the current-52nd Mersenne prime that may very well contain one or two additional Mersenne primes. We are very much looking forward to what they find.

That's enough gushing about prime numbers, so let's get to the biggest math stories of the year that was. Starting with....

The Elephant in the Room

2025 was a year in which avoiding any mention of developments related to Artificial Intelligence (AI) models and the Large Language Models that have come to define the machine learning technology was all but impossible. Hundreds of billions of dollars all around the world are being invested to build and develop the infrastructure needed to support advanced AI systems.

Many disciplines felt the impact of AI technology in 2025 and mathematics is one of many in which AI systems are starting to have a profound impact.

One of the more unique stories we encountered is about one of the first valid mathematical proofs ever produced by a generative AI system. That story came to our attention through a post at X and it's probably best to let that post tell the tale:

GPT-5 just casually did new mathematics.

Sebastien Bubeck gave it an open problem from convex optimization, something humans had only partially solved. GPT-5-Pro sat down, reasoned for 17 minutes, and produced a correct proof improving the known bound from 1/L all the way to 1.5/L.

This wasn’t in the paper. It wasn’t online. It wasn’t memorized. It was new math. Verified by Bubeck himself.

Humans later closed the gap at 1.75/L, but GPT-5 independently advanced the frontier. A machine just contributed original research-level mathematics.

If you’re not completely stunned by this, you’re not paying attention.

We’ve officially entered the era where AI isn’t just learning math, it’s creating it.

Here's the X post in which Bubeck announced the accomplishment.

But as Bubeck later notes, while GPT-5 did something that was both new and novel, humans still beat AI to the punch and delivered a better proof focused on the convex optimization problem that outperforms the AI-generated proof:

Now the only reason why I won't post this as an arxiv note, is that the humans actually beat gpt-5 to the punch :-). Namely the arxiv paper has a v2 https://arxiv.org/pdf/2503.10138v2 with an additional author and they closed the gap completely, showing that 1.75/L is the tight bound.

While that better proof clearly beat its accomplishment, the unexpected artificially generated proof demonstrates AI systems are becoming more capable and useful. Because they are, more things are becoming possible.

Later in the year, a paper revealed another major advancement for the use of AI technologies in maths. Here's that story, which involves events that took place in 2024:

At the 2024 International Mathematical Olympiad (IMO), one competitor did so well that it would have been awarded the Silver Prize, except for one thing: it was an AI system. This was the first time AI had achieved a medal-level performance in the competition's history....

The AI is AlphaProof, a sophisticated program developed by Google DeepMind that learns to solve complex mathematical problems. The achievement at the IMO was impressive enough, but what really makes AlphaProof special is its ability to find and correct errors. While large language models (LLMs) can solve math problems, they often can't guarantee the accuracy of their solutions. There may be hidden flaws in their reasoning.

AlphaProof is different because its answers are always 100% correct. That's because it uses a specialized software environment called Lean (originally developed by Microsoft Research) that acts like a strict teacher verifying every logical step. This means the computer itself verifies answers, so its conclusions are trustworthy....

In addition to solving seemingly intractable math problems, AlphaProof could also be employed by mathematicians to correct their work and help them develop new theories.

The integration of a well-established proof assistant in the AI system represents a major advance and directly addresses one of generative AI technology's main shortcomings: these systems can "hallucinate" and produce absolutely garbage results. Kind of like an editorial cartoon featuring an emcee about to announce the winner of The Biggest Math Story of the Year who has too many hands and fingers.

For mathematicians though, that risk has become much smaller because of the strict discipline imposed by the proof assistant. If the attempted results fail to pass muster according to the logic imposed by the proof assistant, they're rejected.

This is probably the last year in which it might be possible to contain AI-related developments within a single section of a year-end wrap up. Fortunately for humans, none of these stories represent the Biggest Math Story of 2025!

Cracking How a Cryptocurrency Collapsed

In 2022, the cryptocurrency Terracoin collapsed. At the time, there were indications it had been the result of a coordinated effort to crash the cryptocurrency by a handful of traders who stood to profit as other investors got wiped out. But the question of exactly how it was done has been a mystery. Three years later, intrepid researchers using advanced mathematical tools and specialized software they developed cracked the mystery of how it was done. Here's an excerpt:

In a new study published in ACM Transactions on the Web, researchers from Queen Mary University of London have unveiled the intricate mechanisms behind one of the most dramatic collapses in the cryptocurrency world: the downfall of the TerraUSD stablecoin and its associated currency, LUNA. Using advanced mathematical techniques and cutting-edge software, the team has identified suspicious trading patterns that suggest a coordinated attack on the ecosystem, leading to a catastrophic loss of $3.5 billion in value virtually overnight.

The study, led by Dr. Richard Clegg and his team, employs temporal multilayer graph analysis—a sophisticated method for examining complex, interconnected systems over time. This approach allowed the researchers to map the relationships between different cryptocurrencies traded on the Ethereum blockchain, revealing how the TerraUSD stablecoin was destabilized by a series of deliberate, large-scale trades.

Stablecoins like TerraUSD are designed to maintain a steady value, typically pegged to a fiat currency like the US dollar. However, in May 2022, TerraUSD and its sister currency, LUNA, experienced a catastrophic collapse. Dr. Clegg's research sheds light on how this happened, uncovering evidence of a coordinated attack by traders who were betting against the system, a practice known as "shorting."

"What we found was extraordinary," says Dr. Clegg. "On the days leading up to the collapse, we observed highly unnatural trading patterns. Instead of the usual spread of transactions across hundreds of traders, we saw a handful of individuals controlling almost the entire market. These patterns are smoking gun evidence of a deliberate attempt to destabilize the system."

From a trading standpoint, the apparent conspiracy to crash the TerraUSD stablecoin was possible because it was very thinly traded and lacked the liquidity needed to offset the determined efforts by those shorting the cryptocurrency. But more importantly, Clegg's team identified and developed the mathematical tools needed to detect such an effort, which will have real world application as digital currencies like stablecoins become more common.

"These kids today, with their loud music and hula hoops!..."

Every year, Ig Nobel Prizes are awarded to unusual scientific research that, given the subjects involved, make people laugh and then make them think. Occasionally, there's a mathematics category. Here's our contender for next year's Ig Nobel Prizes, which features the first ever results that "explain the physics and mathematics of hula hooping":

"We were specifically interested in what kinds of body motions and shapes could successfully hold the hoop up and what physical requirements and restrictions are involved," explains Leif Ristroph, an associate professor at New York University's Courant Institute of Mathematical Sciences and the senior author of the paper, which appears in the Proceedings of the National Academy of Sciences....

The results showed that the exact form of the gyration motion or the cross-section shape of the body (circle versus ellipse) wasn't a factor in hula hooping.

"In all cases, good twirling motions of the hoop around the body could be set up without any special effort," Ristroph explains.

However, keeping a hoop elevated against gravity for a significant period of time was more difficult, requiring a special "body type"—one with a sloping surface as "hips" to provide the proper angle for pushing up the hoop and a curvy form as a "waist" to hold the hoop in place....

The paper's authors conducted mathematical modeling of these dynamics to derive formulas that explained the results—calculations that could be used for other purposes.

"We were surprised that an activity as popular, fun, and healthy as hula hooping wasn't understood even at a basic physics level," says Ristroph.

"As we made progress on the research, we realized that the math and physics involved are very subtle, and the knowledge gained could be useful in inspiring engineering innovations, harvesting energy from vibrations, and improving robotic positioners and movers used in industrial processing and manufacturing."

Makes you laugh, then makes you think. While this hula hooping story may be a perfect contender for the Ig Nobel Prizes, it's not the Biggest Math Story of 2025. For that story, please scroll down to the next section....

Unifying the Math for Describing How Fluids Behave at All Scales of the Physical Universe

In 1900, David Hilbert identified what he thought would be the 23 biggest challenges mathematics had to offer. 125 years later, over half of those challenges are still waiting to be fully resolved but Hilbert's sixth challenge was very likely cracked in 2025.

That challenge lay in defining a single set of mathematical axioms that could unite the different maths needed to describe how fluid particles behave at different scales. At the smallest "microscopic" level, the motion of individual fluid particles can be described by Newton's laws of motion. Scaling up to the medium "mesascopic" level, how groups of fluid particles behave can be described with the Boltzmann equation. Scaling up once again to the largest "macroscopic" level however takes yet another formulation, with Navier-Stokes equations taking over the work of describing how the fluid itself moves.

Since 1900, a good amount progress toward resolving Hilbert's sixth problem has been made by many contributors, but a major gap remained. In March 2025, three mathematicians, Yu Deng, Zaher Hani, and Xiao Ma posted a preprint paper that filled the remaining gap in the efforts over the past 125 years to create a unified set of math to describe fluid motion at all levels. Quanta Magazine covers the achievement in uniting the math describing the quantum-level (microscopic), medium-level (mesascopic) and large (macroscopic)-scale dynamics of how fluid particles interact in the first third of the following 20-minute video that provides an excellent overview of we think is the biggest math story of the year:

We have more background in our earlier coverage of the accomplishment. At this writing, their work is still awaiting verification, but if Deng, Hani and Ma's work holds, successfully connecting the math of Newton, Boltzmann, Navier, and Stokes across these three scales of reality represents a massive achievement in mathematical theory.

If you watch the full video, Hong Wang and Joshua Zahl's proof of the three-dimensional Kakeya Conjecture (or Needle Problem), a challenge dating back more than a century that revolves around how to determine what the smallest volume is in which a needle can be spun around all possible orientations within a space. Wang and Zahl's remarkable accomplishment is a runner up for our criteria of practical application given its links to geometric measure theory and harmonic analysis.

Ultimately, the question of which achievement would claim the title of being the Biggest Math Story of 2025 came down to recognizing Deng, Hani and Ma's proof represents the culmination of a challenge set out 125 years ago to firmly underpin the physics of how fluids behave with math that works at all scales of the physical universe. It's hard to get much bigger than that in 2025.

Previously on Political Calculations

The Biggest Math Story of the Year is how we've traditionally marked the end of our posting year since 2014. Here are links to our previous editions and our coverage of other math stories during 2024:

• The Biggest Math Story of the Year (2014)
• The Biggest Math Story of 2015
• The Biggest Math Story of 2016
• The Biggest Math Story of 2017
• The Biggest Math Story of 2018
• The Biggest Math Story of 2019
• The Biggest Math Story of 2020
• The Biggest Math Story of 2021
• The Biggest Math Story of 2022
• The Biggest Math Story of 2023
• The Biggest Math Story of 2024
• The Biggest Math Story of 2025
• The Next Mersenne Prime
• Conway's Game of Love
• Game Theory and Settling the Debts of the Deceased in Ancient Times
• Cogitata Physico-Mathematica
• Impossible Maths
• What President Trump's "Liberation Day" Tariff Math Really Says
• Breakthrough in Unifying Math to Describe How Fluids Behave at Different Scales
• The Trillion Dollar Equation
• Flipping Burgers on an Infinite Plane
• Outside the Box Thinking: Galileo's Geometry of Dante's Inferno
• A Tetrahedron That Always Lands on the Same Side
• AI Delivers an Unexpected Mathematical Proof
• The Geometry Problem Tha6t Could Have Protected the French Royal Jewels
• Gift Ideas for the Math Lovers You Know

This is Political Calculations' final post for 2025. Thank you visiting, we hope found the stories and analysis we've presented throughout the year to be either thought-provoking, informative, or entertaining. We'll see you again in the New Year, which we'll kick off with our annual tradition of presenting a tool to help you find out what your paycheck will look like in 2026.

Image credit: Microsoft Copilot Designer. Prompt: "An editorial cartoon featuring an emcee opening an envelope to announce the biggest math story of the year".