Using machine learning to solve mathematical problems and to search for examples and counterexamples in pure maths research

Abstract: Our recent research can be generally described as applying state-of-the-art technologies of machine learning to suitable mathematical problems. As to machine learning, we use both reinforcement learning and supervised learning (underpinned by deep learning). As to mathematical problems, we mostly concentrate on knot theory, for two reasons; firstly, we have a positive experience of applying another kind of artificial intelligence (automated reasoning) to knot theory; secondly, examples and counter-examples in knot theory are finite and, typically, not very large, so they are convenient for the computer to work with.

Here are some successful examples of our recent work, which I plan to talk about.

1. Some recent studies used machine learning to untangle knots using Reidemeister moves, but they do not describe in detail how they implemented untangling on the computer. We invested effort into implementing untangling in one clearly defined scenario, and were successful, and made our computer code publicly available. 2. We found counterexamples showing that some recent publications claiming to give new descriptions of realisable Gauss diagrams contain an error. We trained several machine learning agents to recognise realisable Gauss diagrams and noticed that they fail to recognise correctly the same counterexamples which human mathematicians failed to spot. 3. One problem related to (and "almost" equivalent to) recognising the trivial knot is colouring the knot diagram by elements of algebraic structures called quandles (I will define them). We considered, for some types of knot diagrams (including petal diagrams), how supervised learning copes with this problem.

mathematical physicscommutative algebraalgebraic geometryalgebraic topologycombinatoricsdifferential geometrynumber theoryrepresentation theory

Audience: researchers in the topic

Machine Learning Seminar

Series comments: Online machine learning in pure mathematics seminar, typically held on Wednesday. This seminar takes place online via Zoom.

For recordings of past talks and copies of the speaker's slides, please visit the seminar homepage at: kasprzyk.work/seminars/ml.html