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.

Nature cover: AI-guided intution

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

Kazhdan-Lusztig polynomial G2 (v equals 1). Simulation by Joel Gibson

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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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:

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:

  1. Mathematical problems (e.g. in linear algebra and probability theory) whose resolution would assist the design, implementation and understanding of current AI models.
  2. Mathematical problems or results resulting from interpretability of ML models.
  3. 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.

Larissa Fedunik-Hofman