AI aids old maths
Australian experts are working with artificial intelligence to develop fundamentally new problem-solving techniques.
For the first time, computer scientists and mathematicians have used artificial intelligence to help prove or suggest new mathematical theorems in the complex fields of knot theory and representation theory.
Sydney University’s Professor Geordie Williamson, one of the world’s foremost mathematicians, has applied the power of Google DeepMind AI processes to explore conjectures in his field of speciality - representation theory.
His co-authors were from the team of computer scientists behind the first computer program to successfully defeat a world champion in the game of Go, in 2016.
“Problems in mathematics are widely regarded as some of the most intellectually challenging problems out there,” Prof Williamson says.
“While mathematicians have used machine learning to assist in the analysis of complex data sets, this is the first time we have used computers to help us formulate conjectures or suggest possible lines of attack for unproven ideas in mathematics.”
Representation theory is a branch of mathematics that explores higher dimensional space using linear algebra.
While computers have long been used to generate data for experimental mathematics, the task of identifying interesting patterns has relied mainly on the intuition of the mathematicians themselves.
That has now changed.
Professor Williamson used DeepMind’s AI to bring him close to proving an old conjecture about Kazhdan-Lusztig polynomials, which has been unsolved for 40 years. The conjectures concern deep symmetry in higher dimensional algebra.
Oxford researchers then took the process further, discovering a surprising connection between algebraic and geometric invariants of knots, and establishing a completely new theorem in mathematics.
In knot theory, invariants are used to address the problem of distinguishing knots from each other. They also help mathematicians understand properties of knots and how this relates to other branches of mathematics.
While of profound interest in its own right, knot theory also has myriad applications in the physical sciences, from understanding DNA strands, fluid dynamics and the interplay of forces in the Sun’s corona.
“AI is an extraordinary tool. This work is one of the first times it has demonstrated its usefulness for pure mathematicians, like me,” Prof Williamson said.
“Intuition can take us a long way, but AI can help us find connections the human mind might not always easily spot.”
The authors hope that this work can serve as a model for deepening collaboration between fields of mathematics and artificial intelligence to achieve surprising results, leveraging the respective strengths of mathematics and machine learning.
“For me these findings remind us that intelligence is not a single variable, like an IQ number. Intelligence is best thought of as a multi-dimensional space with multiple axes: academic intelligence, emotional intelligence, social intelligence,” Prof Williamson said.
“My hope is that AI can provide another axis of intelligence for us to work with, and that this new axis will deepen our understanding of the mathematical world.”