Today, two Oxford mathematicians András Juhász and Marc Lackenby published a paper in Nature in collaboration with authors from DeepMind and with Geordie Williamson at the University of Sydney. The paper describes a new crossover between the fields of mathematics and machine learning, where tools from machine learning have been used to discover new patterns in mathematics.
The project that Juhász and Lackenby focused on was in the area of knot theory. Knots are just closed curves in 3-dimensional space. They now have a very well-developed theory, created by many mathematicians over the past 120 years. This theory has several distinct sub-branches, including the use of non-Euclidean 3-dimensional geometry and the use of 4-dimensional invariants. In collaboration with DeepMind, Juhász and Lackenby were able to discover new and unexpected connections between the non-Euclidean geometry of knots and their 4-dimensional invariants. This new connection is of real interest to topologists, but what makes it particularly distinctive is its discovery using machine learning, which was able to detect non-linear relationships between knot invariants. Because of the huge number of different knot invariants, it would have been difficult to make this mathematical progress without machine learning.
This new use of machine learning in mathematics is undoubtedly going to be very widely applicable. In addition to the proofs of new theorems in knot theory by Juhász and Lackenby, it was used by Geordie Williamson in collaboration with DeepMind to discover new results in representation theory. It seems likely that it will become a new tool in many mathematicians' toolkits.
András Juhász said: “Pure mathematicians work by formulating conjectures and proving these, resulting in theorems. But where do the conjectures come from? 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".
Marc commented: “It 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”.
You can watch a video of Marc discussing the work below. And on March 8th, the collaborators will be giving a Public Lecture about the work.