The mathematics of déjà vu

“At a workshop a few years ago a veteran television journalist asked me what it is that I do. When I told him he said: ’What is this term dynamical? It sounds completely ungrammatical.’

“That was the first time it occurred to me that the term dynamical is not in common usage. I guess he wanted me to call it dynamic systems,” Professor Wilkinson said.

Dynamical systems is concerned with the motion of objects and spaces and how things evolve over time. It has applications in a wide variety of fields such as physics, biology, chemistry, medicine and economics and can answers such seemingly disparate questions as how we could unmix ideal gases in a box, or describe the existence of exoplanets in nearby solar systems.

In her talk, Professor Wilkinson, who is visiting from the University of Chicago, will explore how her work uses a mathematical concept comparable to déjà vu called 'recurrence'. Recurrence is a simple and yet powerful concept: we say that a motion of a space is recurrent if, given enough time, it eventually returns to its original configuration, allowing for a small amount of error.

“I think dynamical systems initially appealed to me precisely because I liked the idea of watching things evolve over time,” Professor Wilkinson explained. “Perhaps I am easily bored and need stimulation. I like to change, and I like to study change.”

The field was born in the 1870s when mathematicians and physicists started to wonder if the laws of physics meant that a planet on a regular orbit could potentially fly out of the solar system sometime in the future.

Henri Poincare, the French astrophysicist and mathematician, studied this question and discovered there are in fact configurations of the solar system that are highly unstable and can appear quite regular for long periods of time before they spin out of control. 

These basic investigations allowed mathematicians and physicists to start approaching other questions using similar methods. Mathematicians working on dynamical systems now study abstract spaces and very abstract motions on the spaces.

Professor Wilkinson’s expertise in this area of abstract mathematics has allowed her to work with other scientists tackling rather particular practical problems. “I have recently started doing a lot of work with physicists and particle accelerator groups. They’re interested in solving questions such as: how do you design this machine so that you don't have a billion particles escaping the machine and punching a hole through a wall?”

This multi-disciplinary application of this research is part of what led to Professor Wilkinson being approached by the School of Mathematics and Statistics to present this public lecture.

The lectureship was established in honour of famously contrarian fund manager, Dr Simon Marais. He was known for his willingness to back unloved stocks until they rose to their anticipated value, his adventurous spirit and for his keen love of mathematics.

“I am excited to give the inaugural Simon Marais lecture because I am drawn to what I've read about Marais's interests and view of the world. His interests dovetail precisely with the roots of the field I study,” Professor Wilkinson said.

Professor Wilkinson will appear on a panel for the University of Sydney SAGE initiative, discussing her career as a woman in the field of mathematics.

“Probably the most difficult challenge is feeling isolated because so many of the people around me are men. When I was younger this intimidated me a lot. There weren't very many role models to look up to. Fortunately, this is changing.

“I go out of my way as a senior mathematician to try to encourage young women to continue their studies and to have confidence in their abilities. It takes time to become an expert in anything, and what is wonderful about maths is that mathematicians come in as many different personalities and styles as you could imagine,” Professor Wilkinson said.

Register to attend the inaugural Dr Simon Marais lecture: The Mathematics of Déjà vu