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ChatGPT helped to deal with the intermediate fragment, but the final evidence was made by people.
In mathematics, there was a rare story: three researchers proved the famous hypothesis, which almost did not believe even the author. Michel Talamagon’s hypothesis for 30 years described a strange property of chaotic sets of points: even in spaces with a huge number of dimensions among the disorder, simple and ordered figures inevitably appear.
French mathematician Michel Talagran formulated a hypothesis about SUPknown in 1995. Talagran received the Abel Prize of 2024, which is often called the mathematical analogue of the Nobel Prize, but before the publication of the new evidence, he himself doubted that the bold assumption would be correct. After the release of the work, the mathematician called the result sensational and the most outstanding event in his life.
The hypothesis is associated with convex figures. A circle or penagon of a convex: if you connect any two points inside the figure with a straight line, the line will remain completely inside the boundary. And Pac-Man (the character of the old game in the form of a yellow circle with an open mouth, as if a piece was cut out of the circle) is no longer suitable: the segment between the points above and under the open mouth will go beyond the contour.
Convex figures exist not only on the plane. In three dimensions, the example is the tetrahedron, and Talagran was interested in spaces with hundreds, billions and even greater dimensions. The idea sounds abstract, but calculations with many parameters are the basis for the search, data analysis and the operation of modern artificial intelligence systems. Each parameter can be considered as a separate measurement.
On the plane, an example looks simple: if you draw several points, a convex figure can be drawn around the points, for example, a circle. In any dimension, there is a way to build a convex shell for a set of points, but with the increase in the number of measurements, the task quickly becomes more and more mathematical steps.
Talagran suggested that there is a much simpler path for large-dimensional spaces. In the strongest version of the hypothesis, the complexity of the construction should not grow together with the number of measurements. Even in billions of measurements, you can find a simple convex shape that covers a significant part of the points.
For specialists in the geometry of spaces of large dimension, such a statement seemed almost impossible. Talagra himself considered the hypothesis rather a challenge and waited for a counter-examplation: a set of points for which a simple convex form will not work. The mathematician for years talked about the problem at lectures and even promised $ 2 000 for solving this and related problem, but no one received the award.
The breakthrough was outlined last summer, when the mathematician of the California Institute of Technology Antoine Song translated the geometric task into the language of probability theory. Instead of talking about convex figures, the task turned into a statement about the random choice of points in space according to certain statistical rules.
A new look made the problem closer to solving. After the Son-in-Printon report in December, the mathematician Assaf Naor, who did not participate in the work, expected full proof to appear soon. However, in the discussion remained the missing fragment associated with a mathematical object, with which Sleep had not encountered before.
At this stage, Son and his student Dunming, or Merrick, Hua turned to ChatGPT. After clarifying queries, the large language model helped to deal with the right statement and offered proof of the intermediate result. Later, the mathematician of Princeton University Stefan Tudose joined the work, who knew the necessary object well and independently prepared his own proof.
Sleep and Hua decided to use the proof of Tudoso: the option was more general and meaningful. Later, researchers found old publications with ideas similar to the answer ChatGPT, so the originality of the model’s contribution can not be verified. The final work directly mentions the use of a large language model, but the mathematical result does not relied on the proof proposed by AI.
ChatGPT did not co-author of the proof and did not solve the hypothesis instead of mathematicians. The model helped Sona and Hua faster to go through an unfamiliar site, but the final work included proof of Tudos. At the same time, history shows that AI tools are increasingly helping mathematicians navigate in foreign areas and quickly find the necessary ideas in complex literature.
The full implications of the evidence are not yet clear. The new link between geometry and probability theory can affect the methods of working with data of large dimension, including machine learning tasks and analyzing complex sets of parameters. Talaran himself has already combined his previous bets into a regular award, which will be first awarded in 2032 or after his death, if it happens earlier.
In mathematics, there was a rare story: three researchers proved the famous hypothesis, which almost did not believe even the author. Michel Talamagon’s hypothesis for 30 years described a strange property of chaotic sets of points: even in spaces with a huge number of dimensions among the disorder, simple and ordered figures inevitably appear.
French mathematician Michel Talagran formulated a hypothesis about SUPknown in 1995. Talagran received the Abel Prize of 2024, which is often called the mathematical analogue of the Nobel Prize, but before the publication of the new evidence, he himself doubted that the bold assumption would be correct. After the release of the work, the mathematician called the result sensational and the most outstanding event in his life.
The hypothesis is associated with convex figures. A circle or penagon of a convex: if you connect any two points inside the figure with a straight line, the line will remain completely inside the boundary. And Pac-Man (the character of the old game in the form of a yellow circle with an open mouth, as if a piece was cut out of the circle) is no longer suitable: the segment between the points above and under the open mouth will go beyond the contour.
Convex figures exist not only on the plane. In three dimensions, the example is the tetrahedron, and Talagran was interested in spaces with hundreds, billions and even greater dimensions. The idea sounds abstract, but calculations with many parameters are the basis for the search, data analysis and the operation of modern artificial intelligence systems. Each parameter can be considered as a separate measurement.
On the plane, an example looks simple: if you draw several points, a convex figure can be drawn around the points, for example, a circle. In any dimension, there is a way to build a convex shell for a set of points, but with the increase in the number of measurements, the task quickly becomes more and more mathematical steps.
Talagran suggested that there is a much simpler path for large-dimensional spaces. In the strongest version of the hypothesis, the complexity of the construction should not grow together with the number of measurements. Even in billions of measurements, you can find a simple convex shape that covers a significant part of the points.
For specialists in the geometry of spaces of large dimension, such a statement seemed almost impossible. Talagra himself considered the hypothesis rather a challenge and waited for a counter-examplation: a set of points for which a simple convex form will not work. The mathematician for years talked about the problem at lectures and even promised $ 2 000 for solving this and related problem, but no one received the award.
The breakthrough was outlined last summer, when the mathematician of the California Institute of Technology Antoine Song translated the geometric task into the language of probability theory. Instead of talking about convex figures, the task turned into a statement about the random choice of points in space according to certain statistical rules.
A new look made the problem closer to solving. After the Son-in-Printon report in December, the mathematician Assaf Naor, who did not participate in the work, expected full proof to appear soon. However, in the discussion remained the missing fragment associated with a mathematical object, with which Sleep had not encountered before.
At this stage, Son and his student Dunming, or Merrick, Hua turned to ChatGPT. After clarifying queries, the large language model helped to deal with the right statement and offered proof of the intermediate result. Later, the mathematician of Princeton University Stefan Tudose joined the work, who knew the necessary object well and independently prepared his own proof.
Sleep and Hua decided to use the proof of Tudoso: the option was more general and meaningful. Later, researchers found old publications with ideas similar to the answer ChatGPT, so the originality of the model’s contribution can not be verified. The final work directly mentions the use of a large language model, but the mathematical result does not relied on the proof proposed by AI.
ChatGPT did not co-author of the proof and did not solve the hypothesis instead of mathematicians. The model helped Sona and Hua faster to go through an unfamiliar site, but the final work included proof of Tudos. At the same time, history shows that AI tools are increasingly helping mathematicians navigate in foreign areas and quickly find the necessary ideas in complex literature.
The full implications of the evidence are not yet clear. The new link between geometry and probability theory can affect the methods of working with data of large dimension, including machine learning tasks and analyzing complex sets of parameters. Talaran himself has already combined his previous bets into a regular award, which will be first awarded in 2032 or after his death, if it happens earlier.
