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.
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.
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.
The shape of a mug can stretch and morph into the shape of a donut without cutting or adding any pieces. For that reason, a mug and a donut can be members of the same topological family. Lucas Vieira/Wikimedia Commons (Public Domain)
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.
A Möbius strip is a curious shape with no set inside or outside. A ball would be able to roll over the entire surface of the strip without having to jump from one side to the other. This is one example of a famous shape in topology.PeterHermesFurian/iStock/Getty Images Plus
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.
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.
Claude Monet painted Impression, Sunrise (Impression, soleil) in 1872, unknowingly giving a name to an art style and movement that science says contains an underlying mathematical pattern. Image credit: Musée Marmottan Monet, Paris.
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.
Sunset at Montmajour is a landscape in oils painted by the Dutch artist Vincent van Gogh on 4 July 1888, but was not authenticated until 2013. By Vincent van Gogh – 1. twitter.com2. nrc.nl, image, Public Domain, https://commons.wikimedia.org/w/index.php?curid=28169507
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.”
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.
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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