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New mathematical theorems for representation and knot theory

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A startling relationship between algebraic and geometric invariants of knots has been discovered, leading to the discovery of an entirely new mathematical theorem.

For the first time, artificial intelligence has been applied by computer scientists and mathematicians to prove or suggest new mathematical theorems in the complicated domains of knot theory and representation theory.

The incredible findings were published today in Nature.

Professor Geordie Williamson is the Director of the University of Sydney Mathematical Research Institute and a leading mathematician in the world. He used the capabilities of Deep Mind’s AI processes to examine conjectures in his field of specialization, representation theory, as a co-author of the study.

“Problems in mathematics are widely regarded as some of the most intellectually challenging problems out there,: said Professor Williamson.

“While mathematicians have used machine learning to assist in the analysis of complex data sets, this is the first time we have used computers to help us formulate conjectures or suggest possible lines of attack for unproven ideas in mathematics.”

Proving mathematical conjectures

“Working to prove or disprove longstanding conjectures in my field involves the consideration of, at times, infinite space and hugely complex sets of equations across multiple dimensions,” said Professor Williamson.

While computers have long been used to create data for experimental mathematics, the challenge of detecting interesting patterns has mostly relied on mathematicians’ intuition.

That has now changed.

Professor Williamson used DeepMind’s artificial intelligence to get close to proving a 40-year-old conjecture regarding Kazhdan-Lusztig polynomials. Deep symmetry in higher-dimensional algebra is the subject of the conjectures.

Professor Marc Lackeby and Professor András Juhász of the University of Oxford, who co-authored the paper, have taken the procedure a step further. They uncovered an unexpected link between algebraic and geometric invariants of knots, resulting in an entirely new mathematical theory.

Invariants are employed in knot theory to solve the challenge of recognizing knots from one another. They also assist mathematicians in comprehending knot qualities and how they connect to other disciplines of mathematics.

While knot theory is fascinating in and of itself, it also has a wide range of applications in the physical sciences, including comprehending DNA strands, fluid dynamics, and the interaction of forces in the Sun’s corona.

“Pure mathematicians work by formulating conjectures and proving these, resulting in theorems. But where do the conjectures come from?,” said Professor Juhász.

“We have demonstrated that, when guided by mathematical intuition, machine learning provides a powerful framework that can uncover interesting and provable conjectures in areas where a large amount of data is available, or where the objects are too large to study with classical methods.”

“Intuition can take us a long way, but AI can help us find connections the human mind might not always easily spot,” said Professor Williamson.

The authors think that by combining the skills of mathematics and machine learning, this study can serve as a model for strengthening collaboration between the fields of mathematics and artificial intelligence to generate remarkable outcomes.

“For me these findings remind us that intelligence is not a single variable, like an IQ number. Intelligence is best thought of as a multi-dimensional space with multiple axes: academic intelligence, emotional intelligence, social intelligence,” said Professor Williamson.

“My hope is that AI can provide another axis of intelligence for us to work with, and that this new axis will deepen our understanding of the mathematical world.”

Source: 10.1038/s41586-021-04086-x

Image Credit: Getty

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