高清福利片

For the first time, computer scientists and mathematicians have used artificial intelligence to help prove or suggest new mathematical theorems in the complex fields of knot theory and representation theory.

The astonishing results have been published today in the pre-eminent scientific journal, .

Professor Geordie Williamson is Director of the and one of the world鈥檚 foremost mathematicians. As a co-author of the paper, he applied the power of Deep Mind鈥檚 AI processes to explore conjectures in his field of speciality, representation theory.

His co-authors were from DeepMind - the team of computer scientists behind , the computer program successfully in the game of Go in 2016.

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

鈥淲hile 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.鈥�

Proving mathematical conjectures

Professor Williamson is a globally recognised leader in representation theory, the branch of mathematics that explores higher dimensional space using linear algebra.

In 2018 he was elected the聽youngest living Fellow of the Royal Society聽in London, the world鈥檚 oldest and arguably most prestigious scientific association.

鈥淲orking 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,鈥� Professor Williamson said.

While computers have long been used to generate data for experimental mathematics, the task of identifying interesting patterns has relied mainly on the intuition of the mathematicians themselves.

That has now changed.

Professor Williamson used DeepMind鈥檚 AI to bring him close to proving an old conjecture about Kazhdan-Lusztig polynomials, which has been unsolved for 40 years. The conjectures concern deep symmetry in higher dimensional algebra.

颁辞-补耻迟丑辞谤蝉听听补苍诲听聽from the University of Oxford have taken the process a step further. They discovered a surprising connection between algebraic and geometric invariants of knots, establishing a completely new theorem in mathematics.

In knot theory, invariants are used to address the problem of distinguishing knots from each other. They also聽help mathematicians understand properties of knots and how this relates to other branches of mathematics.

While of profound interest in its own right, knot theory also has myriad applications in the physical sciences, from understanding DNA strands, fluid dynamics and the interplay of forces in the Sun鈥檚 corona.

Human intuition meets machine

Professor Juh谩sz said: 鈥淧ure mathematicians work by formulating conjectures and proving these, resulting in theorems. But where do the conjectures come from?

鈥淲e have demonstrated that, when guided by mathematical intuition, machine learning provides a powerful framework聽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."

Professor Lackeby said: 鈥淚t has been fascinating to use machine learning to discover new and unexpected connections between different areas of mathematics. I believe that the work that we have done in Oxford and in Sydney in collaboration with DeepMind demonstrates that machine learning can be a genuinely useful tool in mathematical research.鈥�

Lead author from DeepMind,聽, said: 鈥淲e think AI techniques are already sufficiently advanced to have an impact in accelerating scientific progress across many different disciplines. Pure maths is one example and we hope that this聽Nature聽paper can inspire other researchers to consider the potential for AI as a useful tool in the field."聽

Professor Williamson said: 鈥淎I is an extraordinary tool. This work is one of the first times it has demonstrated its usefulness for pure mathematicians, like me.

鈥淚ntuition can take us a long way, but AI can help us find connections the human mind might not always easily spot.鈥�

The authors hope that this work can serve as a model for deepening collaboration between fields of mathematics and artificial intelligence to achieve surprising results, leveraging the respective strengths of mathematics and machine learning.

鈥淔or 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,鈥� Professor Williamson said.

鈥淢y 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.鈥�

Declaration

This research was funded by DeepMind, which is owned by Alphabet Inc, the parent company of Google.