The New York Post wrote that OpenAI "stunned the mathematics world" last week when it announced its AI model "cracked a legendary math problem left unsolved for nearly a century in just 88 hours." But a New York University professor "has come forward alleging OpenAI may have copied his work after asking him to collaborate — with one of the company's scientists allegedly warning he might 'ruin his career' if he refused." [P]rofessor Tristan Buckmaster, a British-Australian researcher who trained in Germany... had been working with Long Island-raised math prodigy Levent Alpöge, a researcher with OpenAI rival Anthropic, on a different elusive fluid dynamics equation, known as the Euler problem, when he found a solution using AI tools from both Anthropic and ChatGPT. He said scientists from OpenAI approached him and tried to convince him to jointly announce the solution, claiming they had independently solved Navier-Stokes. Buckmaster questioned the timing of their finding the solution in a blog post, noting OpenAI had only done so "in the past few days" — after the company became aware of his work. He asked OpenAI experts whether its AI systems used to solve the equation had been "trained on, or had access to" his research, but said they declined to answer, prompting him to wonder how they had advanced toward the solution so quickly. The puzzle is "not the direction one arrives at in a few days," Buckmaster wrote incredulously... In OpenAI's post on its website sharing the Navier-Stokes proof, which it credited to Astra, the company's latest AI bot, the firm insisted no user data was incorporated into its model to solve the problem. However, it noted that it "cannot rule out" that anonymized data was used to "improve our models." "We congratulate Levent Alpöge and Tristan Buckmaster on their remarkable mathematical work," OpenAI said in the announcement. Buckmaster shot back, "Is it ethical to use customer's data to try to scoop their customer?" Alpöge's affiliation with Anthropic "seems to have been a sore point for OpenAI," writes TechCrunch. Buckmaster's public statement says the OpenAI's Sebastien "twice asserted that he wanted Levent removed from authorship" and also said that "it was so annoying that Levent works at Anthropic" — but that Buckmaster still refused to remove the Anthropic mathematician's credit. I said that if OpenAI released its result in the way proposed I would go public with what happened. The reply was, "Why would you ruin your career?" [OpenAI's Bubeck said later on X that he'd meant unfounded accusations could damage Buckmaster's career.] I replied that I am an academic, and asked why he thought going public would ruin my career. The reply was, "If you don't want me to be nice, then I don't have to be nice." Buckmaster later told Australia's public broadcaster ABC that the companies building AI "have zero respect for the scientific community. I mean, it's appalling, honestly." But he went on to say AI has made the race to publish mathematical breakthroughs pointless. "I think it's pointless. Like, I think the game is up...." Professor Buckmaster said he and Dr Alpöge had more research they could publish but questioned the point of racing ahead with it when AI could perform work that had previously occupied researchers for years. He said there needed to be a discussion about how mathematics should operate in this new environment. Professor Buckmaster said some mathematicians had stopped publicly sharing what they were working on to avoid tipping off AI companies about their plans. He has called on academics and people from OpenAI, Anthropic and DeepMind to discuss ground rules for how they worked together. He said the companies' power carried responsibilities that went beyond competing to solve problems before their rivals... Building on the work of mathematicians Diego Córdoba and Luis Martínez-Zoroa, the pair found a solution to the Euler problem, a stepping stone to the Navier-Stokes problem. Professor Buckmaster said he spent weeks checking and improving the "slop" proofs generated by AI... For now, Professor Buckmaster said he wanted to finish existing work and support his PhD students and postdoctoral researchers, rather than chase another Millennium Prize. "I think it's more important to re-evaluate what math is," he said. "The New York University professor also warned Australia about the centralisation of resources into the hands of a small number of privately owned foreign companies..."

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Games may be life with all the hard bits removed, but they provide a way to study why people make the choices they make. Traditional games are usually played against a static background: the rewards per outcome are constant. That limits their relevance to behavior because, in real life, the rewards and consequences of strategic choices are ever-changing. Now, researchers have used a mathematical model to study a series of games that include evolving strategies and randomly varying returns.

A bit of history

Perhaps the most famous game-theory contest is the prisoner’s dilemma. In the prisoner’s dilemma, a pair of thieves have been captured and are being separately interrogated by the police. If both clam up, they will be punished for a lesser crime. If one prisoner makes a deal (defects) then that prisoner gets to go free and the other gets a heavier sentence. If both make a deal, they both get an in-between punishment.

The person running the game can start it with different rewards for cooperating and defecting to explore how the optimum strategy varies with reward and risk, which the players can figure out by varying the strategies across multiple rounds. Depending on the balance between the reward for staying silent (cooperating) and betrayal, the game stabilizes with everyone betraying everyone. In this simple situation, everyone loses.

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OpenAI is at the center of another controversy over machines tackling big math problems. This one’s about the twin prime conjecture.

Twenty-five years after the September 11 terrorist attacks, researchers at the Icahn School of Medicine at Mount Sinai have uncovered evidence that the environments in which World Trade Center responders live today may be quietly shaping their risk of frailty, post-traumatic stress disorder, and depression. The study, published in Nature Communications, is the first to examine how neighborhood-level environmental exposures later in life may interact with the chemical and psychological injuries sustained during the disaster response, adding a new dimension to one of the most intensively studied disaster health cohorts in the world.

The research team analyzed data from 18,734 responders who both live in New York City and receive care through one of the World Trade Center Health Program Clinical Centers of Excellence. These responders represent a population that experienced an extraordinary confluence of toxic exposures—dust, combustion products, airborne particulates, and a cocktail of known and suspected carcinogens—alongside severe psychological trauma. While decades of prior research have documented elevated rates of respiratory disease, cancer, and mental illness in this group, far less attention had been paid to whether the everyday conditions of their neighborhoods might compound or buffer those effects as responders age.

To answer that question, the researchers linked each responder’s residential address to annual estimates of three environmental factors: air pollution, high ambient temperatures, and neighborhood greenness, the last typically quantified through satellite-derived vegetation indices. This exposure-assessment approach, common in the field of environmental epidemiology, allowed the team to characterize the conditions each participant encountered where they lived year after year, rather than relying on broad regional averages. The researchers then examined whether those exposures were associated with the probability of being classified as frail, and with diagnosed PTSD and depression.

The results were striking in their consistency. Responders living in hotter neighborhoods—areas that experience more extreme heat days, a burden that falls disproportionately on densely built urban districts with impervious surfaces and little tree canopy—showed an elevated risk of frailty, a geriatric syndrome marked by diminished physiological reserve and heightened vulnerability to stressors. Elevated concentrations of air pollution were likewise associated with increased risks of both frailty and adverse mental health outcomes. Frailty itself is increasingly understood as a whole-body phenomenon, involving chronic inflammation, immune dysregulation, and cumulative cellular wear, all of which are plausible biological pathways through which heat and pollution could accelerate aging.

The mental health findings carried equal weight. Higher pollution and heat exposure were associated with increased odds of PTSD and depression, while responders living in greener surroundings—those with more parks, trees, and vegetated land near their homes—showed lower risks of all three conditions. This inverse association with greenness aligns with a substantial body of literature suggesting that access to green space can reduce psychological stress, encourage physical activity, dampen inflammation, and moderate urban heat, though the observational design of the study means that causation cannot be definitively established.

Perhaps the most provocative finding concerned interaction effects. The researchers found that responders who had experienced higher levels of World Trade Center–related chemical and psychological exposure appeared more vulnerable to the effects of environmental stressors in their current neighborhoods. In other words, the disaster and the decades that followed may not be independent chapters in these individuals’ health histories. Instead, responders already carrying the biological imprint of trauma and toxic dust—including altered inflammatory profiles, epigenetic changes, and persistent hypothalamic-pituitary-adrenal axis dysregulation—may enter later life with less reserve to absorb additional environmental insults. Everyday exposures that might be merely uncomfortable for a healthy middle-aged adult could, for these individuals, translate into measurable acceleration of physical decline and psychological deterioration.

“Twenty-five years after 9/11, we are still learning about the factors that influence the health of those who responded to the attacks,” said Helena Furman Krasnov, PhD, postdoctoral researcher at the Icahn School of Medicine and the study’s first author. “Our findings suggest that everyday exposures, including heat, air pollution, and access to green space, may contribute to responders’ long-term physical and psychological well-being. Understanding these factors could help identify new opportunities to support healthy aging in this population.”

The study’s senior and corresponding author, Maayan Yitshak-Sade, MPH, PhD, Associate Professor of Environmental Medicine at Mount Sinai and a member of the Institute for Exposomic Research, framed the work within the broader exposome paradigm—the effort to capture the totality of environmental exposures an individual accumulates over a lifetime. “Our findings suggest that the health consequences of 9/11 may not end with the disaster itself,” she said. “For responders already carrying the biological effects of trauma and toxic exposures, everyday environmental conditions may represent an additional source of stress on the body and mind. These findings highlight the importance of considering where people live, not just what they experienced during the disaster.”

That framing underscores a conceptual shift in disaster medicine. Traditional models tend to treat a catastrophic event as a discrete exposure with a defined aftermath. The Mount Sinai findings instead suggest that disaster-related health effects may unfold across an entire life course, modified by conditions that have no obvious connection to the original event. A responder’s risk of frailty in 2026 may reflect not only the hours spent at Ground Zero, but also the temperature of the summers they have endured since, the particulate matter drifting through their streets, and whether a leafy park sits within walking distance of their apartment.

The study arrives at a moment when urban environmental health has become a pressing policy concern. Extreme heat now kills more Americans annually than any other weather phenomenon, and cities including New York have launched ambitious initiatives to expand tree canopy, create cool corridors, and retrofit public housing with resilient cooling infrastructure. Air pollution, even at levels within regulatory limits, continues to be linked to cardiovascular, respiratory, and neurocognitive harm. If the associations identified in this cohort hold up under further scrutiny, they could strengthen the case for targeted environmental interventions—not as amenities, but as clinical supports for a vulnerable population.

The implications also extend well beyond the World Trade Center community. The researchers point out that survivors and responders from other disasters—wildfires that blanket entire regions in smoke and ash, industrial accidents that release toxic plumes, building collapses that bury rescue workers in dust—may face the same dual burden of physical and psychological exposures that continues to shape health long after the initial event. Climate change is intensifying heat waves and wildfire seasons simultaneously, meaning that disaster-affected populations may increasingly find themselves living in precisely the kinds of hostile urban microclimates the study identifies as risky. The framework developed here could inform how health surveillance programs for disaster cohorts are designed, prompting them to collect residential and environmental data alongside clinical outcomes.

The authors are careful to note the study’s observational nature. Associations between neighborhood environment and health can be confounded by socioeconomic status, housing quality, access to care, and the self-selection of healthier individuals into greener, quieter neighborhoods. The investigators controlled for available covariates, and the sheer size of the cohort lends statistical power, but randomized evidence on green space exposure is practically impossible to generate, and residual confounding remains an inherent limitation of the design. Longitudinal follow-up, and ideally natural-experiment analyses—such as the addition of new parks or major infrastructure changes—will be needed to firm up causal inference.

That work is already on the horizon. The team plans to investigate whether beneficial environmental factors, including green space and access to healthy food, can actively reduce the long-term health effects associated with World Trade Center–related exposures—a question that shifts the framing from risk identification to potential intervention. If confirmatory studies support the protective role of greenness, the findings could justify clinical guidance for responders, and potentially for other disaster-exposed populations, that goes beyond medication and monitoring to encompass the places where patients live.

For a population that has already given so much and borne so much, the idea that healing might partly depend on the landscape outside the front door is both sobering and hopeful. It suggests that some of the damage inflicted on September 11, 2001 may not be fixed, but modifiable—and that the next chapter of care for responders could be written not only in clinics, but in city planning offices, park commissions, and the slow, patient work of greening the neighborhoods where aging heroes live out their lives.

For decades, researchers wondered if the Navier-Stokes equations, which describe fluid flow, break down. OpenAI put thousands of agents on the question.

An anonymous reader quotes a report from The New York Times: OpenAIsaid on Tuesday that its newest artificial intelligence technology had solved one of the "Millennium Problems," a collection of important unanswered math questions meant to push the world's leading mathematicians to new heights. The announcement is an another clear sign that A.I. is fundamentally changing the upper reaches of mathematics, which have long been viewed as a pinnacle of human achievement. The change has excited some mathematicians, while stirring concern among others. Over the past year, A.I. systems successfully solved a wide range of problems that have bedeviled mathematicians for decades. But these problems were not as complex, nor as closely watched, as the one that OpenAI's technology has solved over the past several days. The Millennium Problems are among the most heavily researched in the field. "This is a spectacular culmination of the arc we have seen over the past twelve months," OpenAI researcher Sebastien Bubeck said of the company's new solution. The company announced that one of its latest models, which has not yet been released to the public, needed just 88 hours to solve what mathematicians call "the Navier-Stokes existence and smoothness problem." This problem involves a series of equations that are often used to predict the weather. The equations describe the movement of water and other fluids. The Navier-Stokes problem, which has no clear practical value, asks whether these equations completely break down in certain situations. OpenAI's proof claims to have defined just such a situation. This would imply, at least theoretically, that the laws of physics themselves would break down under certain conditions: that, for example, water could be made to spontaneously explode. But mathematicians and physicists do not believe that this mathematical breakdown could really lead to such an outcome in the physical world. The Navier-Stokes problem was one of seven "Millennium Problems" selected by the Clay Mathematics Institute in the year 2000 as a way of tracking the progress of mathematics in the new millennium. The institute, founded by an American businessman named Landon T. Clay, offered a million dollars for the first correct solution to each problem. Before OpenAI's announcement, only one of the problems had been solved. OpenAI says it managed to solve the Navier-Stokes problem by coordinating as many as 10,000 AI agents, an extremely expensive process that may have cost millions of dollars in computing power. For those curious about the findings, the company has released a paper (PDF) and formal Lean proof.

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Are computers in mathematics rewriting the definitions of stupid and smart?

The post The Elegance of a Stupid Question appeared first on Nautilus.

Dark energy might be getting weaker. Scientists are wondering if interactions with dark matter in a “dark dimension” may be responsible.

Symmetry (noun, “SIM-meh-tree”)

A shape with symmetry has parts that match each other in some way.

Imagine cutting a heart from a piece of paper. To ensure the two halves match perfectly, you might fold your paper in half. Then, you can cut both sides of the heart at once. When you unfold the paper, the two sides are identical. The heart is symmetrical.

The line down the center of the heart, left over from folding your paper, is called the line of symmetry. Folding a shape along a line of symmetry matches up the parts. This kind of symmetry, across one line down the middle, is called mirror symmetry. That’s because the two sides of the heart are shaped like mirror reflections of one another.

But symmetry can come in different forms. Rotational symmetry is another example. This describes a shape that you can spin into a new position that looks identical to the first one. A propeller often displays this kind of symmetry.

Symmetry appears in nature. The elegantly branching crystal shards of a snowflake often have rotational symmetry. Leaves of a tree may display mirror symmetry. The body shape of a cat or dog or person also shows this kind of symmetry. When referring to living things, biologists often use the word bilateral symmetry. It means the same thing as mirror symmetry.

Radial symmetry is another example of symmetry in body shapes. This describes a body shape where you can draw several different lines of symmetry that cross the center. Jellyfish and starfish are examples of creatures with radial symmetry.

Not all living things are symmetrical. A marine sponge usually lacks matching parts. So, this type of body plan is described as asymmetrical.

In a sentence

Floral designs on ancient pottery suggest people understood symmetry 8,000 years ago.

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Never mind the “heat death of the universe.” Entropy is just a bridge between two levels of reality.

Never mind the “heat death of the universe.” Entropy is just a bridge between two levels of reality.

The sum is hard to comprehend

The post What Is a Trillion, Really? appeared first on Nautilus.

Topology (noun, “top-AH-low-jee”)

Topology is a field of math that focuses on how molding or stretching a shape alters the space it takes up.

In a way, topology is similar to geometry. Both fields study shapes. However, geometry focuses on rigid measurements, such as the length of lines in a triangle. It answers questions like: If I change the length of this line, how will the other lines change? In geometry, the new triangle is considered a different triangle altogether.

But topology does not focus on such rigid measurements. Instead, topology treats the study of shapes like a game. As in a game, a topologist sets out certain rules. Then, they explore what shapes are possible within those rules.

For instance, a rule might be that all the shapes in a certain family can morph into one another without tearing or gluing parts together. You could explore this yourself by squeezing a water balloon. As you watch the balloon warp, you see continuous changes to its shape. Squeezing the middle causes the outsides to bulge. But the balloon doesn’t break or form new connections. So, all the shapes the balloon takes still obey the rule.

A mathematician might say that all the shapes of this balloon belong to the same topological group.

Another rule might be that all the shapes in a group must have one hole. A donut is an obvious member of this group. But so is a mug. If you made both of these shapes out of clay, you could remold one into the other without tearing the clay or attaching any new pieces. Under the one-hole rule, they are in the same topological family.

A particularly mind-twisty rule might be that all shapes in a family have “non-orientable surfaces.” That means the shape doesn’t have a clear outside and inside. A loop with a twist in it, called a Möbius strip, is a famous example.

Topology can be represented in many ways. Computer models can render and play with shapes. The math of algebra can describe concepts in topology, too.

Setting up rules for shapes and playing around with them is more useful than you might think. It helps scientists understand what is physically possible and impossible. Engineers can use these insights to imagine new technologies. Cosmologists use topology to describe the shape of our universe. And physicists who study string theory use it to probe the nature of matter.

In a sentence

A Möbius strip is one famous example of a curiously mind-bending shape from the field of topology.

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As World Cup action kicks off, we look at the physics of the beautiful game.

Researchers from the Université Polytechnique Hauts-de-France have revealed a technique that successfully analyzed several Vincent van Gogh artworks at a microscopic level to correctly identify well-known, previously identified forgeries from recently authenticated ones

The research team behind the pioneering, non-invasive art authentication process said their approach could provide museums, collectors, and art auction houses with a previously unavailable tool with a proven, scientifically based track record for identifying fake or counterfeit art.

Using Microscopic-Level Surface Scans to Foil Art Forgery

According to a press release announcing the successful counterfeit identification tests, art forgery is an increasingly complex problem. However, the traditional approach of expert opinion combined with historical research and invasive pigment analysis has not always proven successful, with the study authors noting that such methods are “often inconclusive,” leaving museums and collectors to shoulder most of the risk for difficult-to-authenticate works.

More recently, collectors and auctioneers have turned to science for answers, including a report showing that AI may have identified as many as 40 counterfeit artworks sold on online marketplaces. Although no counterfeit versions of Van Gogh’s works were found, that same effort identified potentially fake versions of paintings by Monet and Renoir that were indistinguishable from authentic ones.

In 2023, researchers at the University of Sydney announced the development of a cutting-edge microscope capable of surpassing the diffraction limit. In that story, study co-author Professor Boris Kuhlmey noted that the method, which is ideal for identifying microchip irregularities, “could even be used to reveal hidden layers in artwork, perhaps proving useful in uncovering art forgery or hidden works.”

Last month, The Debrief reported the discovery of a mathematical pattern in many famous expressionist artworks, noting that many abstract artists may naturally arrange shapes and patterns in similar ways, even without knowing the mathematics behind them. If correct, this concept could use science and mathematics to separate fake Impressionist works.

Now, the Université Polytechnique Hauts-de-France team says their new approach takes modern technology a step further, resulting in an unprecedented level of accuracy.

“Fractal Analysis Gives Us a Measurable Fingerprint of an Artist’s Brushwork” 

Unlike the above techniques, the new approach analyzes the texture of a painting’s surface at the microscopic level by converting high-resolution images into three-dimensional surface ‘maps’ of the painting’s upper layers. According to the study authors, the extracted “texture” of a painting allows them to measure an image’s roughness and detail using fractal math. For example, the team said that the subtle patterns of an artist’s brush stroke are so consistent “that they act like a morphological signature unique to that artist.” 

“Fractal analysis gives us a measurable fingerprint of an artist’s brushwork without needing to sample or disturb the painting,” explained Francois Berkmans, the study’s lead researcher.

To test their approach, the researchers scanned several artworks attributed to Renaissance artist Vincent van Gogh. The researchers also used their technique to study the mathematical texture of an artwork called The Plowmen, which the team noted is a “well-documented fake.” Last, they used their method to analyze another Vincent van Gogh work, Sunset at Montmajur, which was authenticated in 2013.

After running their microscopic surface analysis, the team’s new system identified The Plowmen as a “strong outlier” compared to the artist’s other works. Conversely, the new method determined that the two newly authenticated paintings “aligned closely with Van Gogh’s known works.”

New Method “Strengthens” Traditional Analysis as a Complimentary Tool

When discussing the study’s results, Berkmans said their technique “can clearly point out genuine artists and reliably detect known forgeries.” Along with identifying fakes and authentic works, the team said that the new approach successfully separated Van Gogh’s stylistic signatures from those of 17th‑century painter David Klöcker Ehrenstrahl, an achievement they heralded as demonstrating the technology’s “wider potential.” 

Although the new approach had successes, the team said it is not designed to replace current approaches, including complementary analyses such as the chemical examination of materials. Instead, they said their surface-texture approach should be used as a complementary tool to strengthen authentication, resulting in a multifaceted approach that can reduce risk for collectors, museums, and auction houses alike.

“This approach won’t replace traditional expertise, but it significantly strengthens it.”

The study “Preserving Van Gogh’s Painterly Heritage: Topographical and Fractal Insights in Authentication” was published in Surface Topography Metrology and Properties.

Christopher Plain is a Science Fiction and Fantasy novelist and Head Science Writer at The Debrief. Follow and connect with him on X, learn about his books at plainfiction.com, or email him directly at christopher@thedebrief.org.

The result is correct but challenges core norms of mathematics: checking proofs, crediting ideas and keeping research open to everyone.

Physicist Richard Feynman turned a lunch dilemma into a math problem. Researchers finally cracked his notes and found people approximate his solution on their own.

In mid-May, OpenAI announced that an internal AI model had disproved the Erdős unit distance conjecture, a famous problem in discrete geometry that had stumped human mathematicians for the last 80 years.

OpenAI gave several mathematicians early access to the result and published their reactions. Tim Gowers—who won the Fields Medal, the most prestigious prize in mathematics—wrote that “there is no doubt that the solution to the unit-distance problem is a milestone in AI mathematics.”

University of Toronto professor Daniel Litt wrote that “this is the first example of a result produced autonomously by an AI that I find exciting in itself, as opposed to as a leading indicator.”

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These tiny creatures push the limits of how small life can be

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