Dr Tom Oliver, Lecturer in Computer Science and Engineering at the University of Westminster, has spoken to Quanta Magazine about his collaboration with fellow academics and AI that led to the discovery of the Murmuration of Elliptic Curves.
The article emphasises the importance of Elliptic Curves, which Dr Oliver explains are: “The shape defined by a simple equation with variables x and y. You can draw it as a graph just like in school. Although they look easy, they are at the heart of many unsolved problems in theoretical and applied maths. Elliptic curves are used, for example, in bitcoin.”
They hold vital importance in the world of mathematics and have been central to proving many theories throughout history. Due to their significance, the journalist highlights that understanding them is a “high-stakes endeavour”.
The article explains that Dr Oliver and the team’s journey began with a joint interest in machine learning, a type of AI and Computer Science that uses data and algorithms to enable AI to imitate how humans learn, which in turn helps improve its accuracy.
Dr Oliver said: “We decided to do this just to learn what machine learning was, rather than to seriously study mathematics. But we quickly found that you could machine-learn a lot of things.”
Together they ran multiple algorithms through machine learning and discovered they were able to produce accurate results on a variety of investigations. Confused by the accuracy, they set an undergraduate Alexey Pozdnyakov up with the challenge of finding out why their results were so accurate.
Dr Oliver explains: “Alexey Pozdnyakov found an unexpected collective behaviour amongst elliptic curves. No single curve acts that way, it only works for the collections we were using to construct our machine learning experiments. The oscillations he found mean that some features in the data help to answer the question we were asking on a statistical level.”
On this discovery in the article Dr Oliver says: “I don’t think we would have found it without him. Because the experts traditionally normalize ap to have absolute value 1. But he didn’t normalize them … so the oscillations were very big and visible.”
The article concludes: “The statistical patterns that AI algorithms use to sort elliptic curves by rank exist in a parameter space with hundreds of dimensions — too many for people to sort through in their minds, let alone visualize, Oliver noted. But though machine learning found the hidden oscillations, “only later did we understand them to be the murmurations.”
Dr Oliver’s work directly contributes to the United Nations Sustainable Development Goal (SDG) 17: Partnerships for Goals and 9: Industry, Innovation and Infrastructure. Since 2019, the University of Westminster has used the SDGs holistically to frame strategic decisions to help students and colleagues fulfil their potential and contribute to a more sustainable, equitable and healthier society.
Read the full article in Quanta Magazine.
