Human-AI collaboration can uncover new areas of math where datasets are too large to be understood by mathematicians
AI software worked with mathematicians to successfully develop a theorem about the structure of nodes, but the suggestions of the code were so unintuitive that they were initially discarded. It was only later that it was discovered that they offered invaluable information. The work suggests that AI could reveal new areas of math where large data sets make problems too complex for humans to understand.
Mathematicians have long used computers to do the brutal force work of large calculations, and artificial intelligence is even used to disprove mathematical guesswork. But creating a guess from scratch is a much more complex and nuanced problem.
To refute a guess, all an AI needs to do is analyze a variety of inputs to find a single example that contradicts the idea. Rather, developing a conjecture or proving a theorem requires intuition, skill and the combination of many logical steps.
The British artificial intelligence company DeepMind, part of Google’s parent company Alphabet, has already been successful in beating people in chess and Go games with artificial intelligence and in solving the structures of human proteins. it can provide promising clues to human mathematicians for the development of theorems. This work has led to a conjecture in the field of topology and representation theory and a well-established theorem about the structure of knots.
In contrast to most neural network studies, in which an AI receives a large number of examples and learns to recognize or generate similar inputs, the AI here examines existing mathematical constructs for patterns. According to DeepMind, its AI found both familiar and new patterns and guided human Mathematicians to new discoveries.
Marc Lackenby and András Juhász from the University of Oxford worked with DeepMind on a new theorem about the relationship between algebraic and geometric invariants of knots. Knot theory is the study of knots as found in rope, except that in these models the two ends are joined together. Although the field provides information on how to entangle a string, it also has applications in quantum field theory and non-Euclidean geometry.
DeepMind’s artificial intelligence software received details on the two previously separated components of knot theory, algebraic and geometric, and was asked to look for correlations between them, both direct and complex, subtle and non-intuitive. human mathematicians for analysis and refinement. Some of them turned out to be math already established, while others were completely new.
Lackenby says the AI has identified a number of variables that, combined in complex ways, suggest a correlation between the two previously separated fields. Initially, the team took just the three strongest of these suggested variables and tried to make an estimate.
We’ve spent a lot of time proving that and it turns out it’s not quite right, says Lackenby. But it turns out that the fourth and fifth [AI proposal] also control the company in these very subtle ways. So we would actually have saved ourselves a lot of time if we had taken into account the information from machine learning was telling us at face value. The Machine Learning knew at all times.
After taking these additional variables into account, the team was able to complete the guess and also prove the sentence, We worked in a world where our intuitions were challenged, says Lackenby. We haven’t expects there to be such a clear connection between these algebraic and geometric quantities, so I was very surprised.
Various suggestions from the AI led to possible guesses that proved true for millions of examples, but which fell apart on further investigation. Lackenby believes that AI is far from being able to complete the process of analyzing promising clues and developing conjectures or theorems on its own. but which could be invaluable in getting or guiding people into promising areas of study.
I think that improving intuition is absolutely crucial for the mathematician. Intuition guides us, so anything that can help is a really useful tool, he says.
AI also helped Geordie Williamson of the University of Sydney discover an unproven but successfully tested conjecture in representation theory on more than three million examples.
