The AI ​​system breaks the knot theorem – a learning algorithm that improves theoretical mathematics

Knoten

Computers are common in mathematics, but it is up to humans to make new mathematical assumptions or theorems – this fine art of intuitive mathematics is not considered machine-learning. Now an AI system teaches us better: According to researchers in the journal Nature, it was with his help that mathematicians discovered a new theory in knot theory and a new guess in algebra.

Op Kepler’s assumption For packing of balls, puzzle Prime Twins Or the question of whether Infinity: In mathematics, it is often used to recognize common forms and laws and to prove them mathematically. Until these relationships and laws are yet to be proven, mathematicians will continue to speculate; If they are proven, they are considered a theorem.

The role of intuition

Although computers have long been instrumental in calculating or analyzing large numbers, the establishment of new assumptions and theories is considered a field of human intuition. Recognition of previously unrecognized forms and relationships cannot be programmed and often occur spontaneously. The best example of such intuitive flashes of inspiration is the Indian mathematician Srinivasa Ramanujan, who made countless wonderful conjectures on the theory of numbers – seemingly nowhere else.

“A mathematician’s intuition plays a vital role in mathematical discoveries,” explained Alex Davis and his colleagues at the Google Deep Mind Research Center in London. But what if there is an AI system that works for this mathematical intuition and provides a certain amount of “nourishment”? A team of AI researchers and mathematicians has now shown how this works.

Knot theory as a testing ground

For the study, two teams of mathematicians used the AI ​​system developed by DeepMind to test mathematical hypotheses in two subject areas. Its purpose was to create new conjectures or theories. Andres Juhas, a mathematician at the University of Sydney, explains: “Learning machines can detect interesting and proven guesses in areas with high data or very large objects.

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In particular, he uses adaptation AI to clarify a question from the field of knot theory: “Our hypothesis is that there is still an undiscovered relationship between the hyperbolic and algebraic variants of the knot,” Juhas explains. In this case, variations are mathematical properties that describe one node and differentiate it from the other. The team trained the AI ​​system, check the set of geometric variations to see where possible cross-links might be.

Teamwork creates a new theorem

In fact, the DeepMind algorithm struck gold: it identified three variants that represented the algebraic signature of the terminal and the geometric interpretation of its slopes. Based on this discovery, the mathematician was able to draw a new hypothesis from it and prove it directly.

Teamwork between AI and humans resulted in a new mathematical theorem. “This theorem is the first theorem that combines the algebraic and geometric variations of nodes. It has many interesting applications,” explains the scientists.

“It’s surprising that such a simple and deep connection has not gone unnoticed – in an area that has been seriously researched,” Juhas and his team write.

New conjecture in the theory of representation

DeepMind-KI The second part of mathematics has proven itself as a mathematician in the theory of representation. This part of linear algebra deals with a particular way of drawing and describing elements and groups using symmetries at high dimensional intervals. “In order to make or prove conjectures in my area of ​​expertise, it is sometimes necessary to have infinite gaps and very complex equations in many dimensions,” explains Giordi Williamson of the University of Sydney.

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With the help of the AI ​​system, Williamson and his colleagues succeeded in dispelling speculation about the Kazhdan-Lusztig polynomials that had not been completed for 40 years. “This is the first time AI has proven its usefulness to a purely theoretical mathematician like me,” Williams said. “While intuition takes us far, AI can help us discover connections that the human mind can never see.”

Useful in other areas as well

According to the research team, early examples show that machine learning can be a valuable aid in even the most complex and concise subjects, such as theoretical mathematics. “In our opinion, AI systems have been developed sufficiently to accelerate scientific progress in many fields,” says DeepMind researcher Davis. “Pure mathematics is an example of this.” (Nature, 2021; doi: 10.1038 / s41586-021-04086-x)

Quelle: Nature, University of Sydney

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