Maths and machine learning
Can artificial intelligence help solve humanity’s tough problems? This is one of the most pressing questions in contemporary science.
Over the last two decades, we have seen neural networks starting to perform well on tasks that humans find easy, like image and speech recognition. However, most tasks that require conscious thought, like mathematics problems, are beyond the capacity of current neural networks.
SMRI Director Geordie Williamson and a team of mathematicians from Oxford University have collaborated with AI lab DeepMind, using machine learning to help prove or suggest new mathematical theorems. This is one of the first examples where machine learning has been used to guide human intuition on decades-old problems. The results were published in the preeminent journal Nature in December 2021. Watch our interview with Geordie explaining what happens when mathematical intuition meets AI.
Geordie applied the power of DeepMind’s AI processes to explore conjectures in representation theory, a branch of pure mathematics. This has brought him closer to proving a conjecture concerning deep symmetry in higher dimensional algebra, which has been unsolved for 40 years.
In parallel work, the Oxford mathematicians used the AI to discover surprising connections in the field of knot theory, establishing a completely new mathematical theorem.
AI-assisted research in representation theory
This video shows a visualisation in representation theory—while it doesn’t show machine learning data, it illustrates the insights one can try to learn. Representation theory studies abstract algebra at higher dimensions by representing their elements as linear transformations. This makes it easier to identify symmetries and other patterns deep within their structures.
Geordie’s AI-assisted research focused on Kazhdan-Lusztig (KL) polynomials, which are important measurements within representation theory. A chemical analogy is to describe representation theory as atoms, and KL polynomials as the atomic numbers of mathematical structure. This video shows the development of certain KL polynomials: they quickly develop complicated patterns of symmetry.
Artificial intelligence and machine learning can assist in the discovery of patterns in higher dimensions, revealing patterns faster or unseen by human methods alone. In the project, Geordie focused on a particular conjecture in representation theory (the combinatorial invariance conjecture), which involved associating a KL polynomial with an abstract object called a Bruhat graph. Geordie and DeepMind colleagues trained a neural network to absorb a Bruhat graph and generate a prediction for the KL polynomial, with impressively accurate results.
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- AP Archive video: “Can the latest AI technology out-math the mathematicians?” feat. Geordie Williamson and Lindon Roberts, University of Sydney (22 May 2023)
- SBS News in Depth: “Mathematicians say artificial intelligence just doesn’t add up” feat. Geordie Williamson and Lindon Roberts (20 May 2023)
- SMRI YouTube video (14 March 2022): “Intuition meets AI in pure mathematics”, also featured in Math off the grid Carnival of Mathematics blog post (3 April 2022)
- Quanta magazine article (15 Feb 2022): “Machine Learning Becomes a Mathematical Collaborator”
- University of Sydney news feat. Geordie Williamson (2 Dec 2021): “Mathematicians use DeepMind AI to create new methods…”
- The Conversation article by Geordie Williamson (2 Dec): “Mathematical discoveries take intuition and creativity…”
- Nature News feature (1 Dec): “DeepMind’s AI helps untangle the mathematics of knots”
- DeepMind research blog post (1 Dec): “Exploring the beauty of pure mathematics in novel ways”
- New Scientist technology news article (1 Dec): “DeepMind AI collaborates with humans on two mathematical breakthroughs”
- University of Oxford news release (1 Dec): “Machine learning helps mathematicians make new connections”
- VentureBeat article (1 Dec): “DeepMind claims AI has aided new discoveries and insights in mathematics”
- TechCrunch article (2 Dec): “AI proves a dab hand at pure mathematics and protein hallucination”
- COSMOS article (3 Dec): “The AI making waves in complex mathematics”
- Numberphile podcast feat. Alex Davies, DeepMind & Marcus du Sautoy, University of Oxford (3 Dec): “Google’s ‘DeepMind’ does Mathematics”
- 2ser Science Spotlight with Cameron Furlong feat. Geordie Williamson (9 Dec): “Mathematics and AI”
- Silicon Reckoner blog post (3 Dec): “News flash: DeepMind and “the beauty of pure mathematics”
- Combinatorics and more blog post (4 Dec): “To cheer you up in difficult times 33: Deep learning leads to progress in knot theory and on the conjecture that Kazhdan-Lusztig polynomials are combinatorial”
- Science Alert tech news (4 Dec): “AI Is Discovering Patterns in Pure Mathematics That Have Never Been Seen Before”
- Towards AI Deep Learning blog post (4 Dec): “Inside DeepMind’s New Efforts to Use Deep Learning to Advanced Mathematics”
- DailyAlts Artificial Intelligence news (6 Dec): “Artificial Intelligence: New Revelations In Pure Mathematics By AI”
- Live Science news (7 Dec): “DeepMind cracks ‘knot’ conjecture that bedeviled mathematicians for decades”
- SingularityHub article (7 Dec): “How DeepMind’s AI Helped Crack Two Mathematical Puzzles That Stumped Humans for Decades”
- IANS news (2 Dec): “DeepMind & Mathematicians Use AI To Solve The Knot Problem”
- Analytics Drift data science news (6 Dec): “DeepMind Makes Huge Breakthrough by Discovering New Insights in Mathematics”
- Analytics India Mag opinions piece (7 Dec): “Australian mathematician cuts through knotty questions with AI”
- Ask Innovative India AI news (9 Dec): “DeepMind’s AI aids in the deciphering of knot mathematics!”
- Article in Sky News (2 Dec): “Mathematicians hail breakthrough in using AI to suggest new theorems”
- Article in Independent (2 Dec): “Scientists make huge breakthrough to give AI mathematical capabilities never seen before”
- Article in The Times (2 Dec): “DeepMind’s artificial intelligence software helps mathematicians pinpoint patterns”
Machine learning for the working mathematician (SMRI course: Semester One, 2022)
The Machine Learning for the Working Mathematician course is an introduction to ways in which machine learning (and in particular deep learning) has been used to solve problems in mathematics. The seminar series was organised by Joel Gibson, Georg Gottwald, and Geordie Williamson.
We aim for a toolbox of simple examples, where one can get a grasp on what machine learning can and cannot do. We want to emphasise techniques in machine learning as tools that can be used in mathematics research, rather than a source of problems in themselves. The first six weeks or so will be introductory, and the second six weeks will feature talks from experts on applications.
Two nice examples of recent work that give the ‘flavour’ of the seminar are:
- Advancing mathematics by guiding human intuition with AI, a collaboration of Geordie Williamson (an organiser of MLWM), Mark Lackenby and Andras Juhasz (at the University of Oxford) with the team at Google Deepmind which led them to a new theorem in knot theory and a new conjecture in representation theory. This was recently featured in the 2022 State of AI report.
- Constructions in combinatorics via neural networks, where Adam Zsolt Wagner (invited to speak) uses ML strategies to come up with several counterexamples to conjectures in graph theory and other combinatorial problems.
The lectures can be viewed on the MLWM YouTube playlist, which continues to be a valuable resource for mathematicians interested in machine learning. Jupyter notebooks, lecture notes, references and supplementary material can be found in the detailed MLWM course overview.
Mathematical Challenges in AI (SMRI course: Semester Two, 2023)
The Mathematical challenges in AI seminar series is the successor of Machine Learning for the Working Mathematician.
The main focus of the seminars this year will be to explore the mathematical problems that arise in modern machine learning. For example, we aim to cover:
- Mathematical problems (e.g. in linear algebra and probability theory) whose resolution would assist the design, implementation and understanding of current AI models.
- Mathematical problems or results resulting from interpretability of ML models.
- Mathematical questions posing challenges for AI systems.
Our aim is to attract interested mathematicians to what we see as a fascinating and important source of new research directions.
View the Mathematical challenges in AI YouTube playlist.
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- IMU-SMRI Mathematical Colloquia and Panels: 25–26 March 2024 at the University of Sydney - 2024 March 13