Cracking the code

Often hailed as the backbone of science, mathematics and statistics are thought by many to provide the tools for other sciences. The impact of these disciplines is, however, much greater than this, giving us the language to make new observations and predictions. Mathematicians are changing the way we approach problems and the way we think about the world around us.

“Mathematical models can predict the future from a known set of initial states,” reveals Payne-Scott Professor Nalini Joshi, from the School of Mathematics and Statistics.

“The trouble is that sometimes the initial state has to be known to infinite precision before we can make a prediction. So, we have to invent new methods that have the power to reveal the heart of a problem.”

Mathematical models consist of an interconnected set of equations. A model of a linear system has a profile of equations whose solutions build to give a prediction. It is nonlinear systems that present a barrier to prediction. For instance, waves at a beach collide, break and change in ways that cannot be explained as a sum of their parts.

Nonlinear systems surround us in most of the phenomena in our universe, in questions such as: What transition does a metal undergo when it is heated to a point at which it becomes magnetic? Is a needlelike shape the only possible interface between solid and air for a growing crystal? Each answer relies on describing solutions in difficult limits. A limit is a solution an equation approaches but never reaches.

“I find crucial information about solutions in limits,” explains Joshi.

Her work on intricate limits started with the classical Painlevé equations, which model a gamut of real life situations, ranging from the transition to magnetism in a bar of metal, to queuing times for boarding an airplane.

“These situations have solutions that are at first extremely close but, with the passage of time, become hugely different.”

Outside the world of mathematics, we think about this as the ‘butterfly effect’.

“Mathematicians don’t expect to crack the code and predict the future, but we can use Painlevé equations to illuminate areas of our missing knowledge,” says Joshi.

“Painlevé equations come with an enormous amount of information about the solutions of nonlinear systems. If you have the right tool you can reveal their striking solutions,” explains Joshi.

She aims to branch out and show that these striking solutions also occur in biological systems.

Joshi also explores how we can predict the results of discrete models, like those generated by computer simulations. When a model of the weather is simulated on a computer, the equations that balance pressure, temperature, air velocity, and their continuous rates of change, become equations that are updated in discrete time steps.

“Like a strobe light on a dance floor, we can only grab information at distinct moments in time. This makes prediction difficult as the results can be volatile,” she explains.

Currently, Joshi is obsessed with cellular automata. These describe how the state of a cell changes according to the states of its neighbouring cells.

“They are another way to explore the properties of nature, and the beautiful patterns that flow through natural systems.”

Joshi is generous in offering her time to mentor other academics and students in the School of Mathematics and Statistics to share the power of maths with others.

“I’ve mentored over 30 researchers in my career. Particularly for younger researchers I make the invisible visible by pointing out hidden barriers to success. I encourage leadership skills and independence, so they blaze their own pathway in research.”

She’s also passionate about encouraging females to pursue careers in science, technology, engineering, mathematics and medicine, collectively known as STEMM, and playted an integral part in establishing the Science in Australia Gender Equity (SAGE) program in Australia and was its first co-chair. She currently sits on SAGE’s Expert Advisory Group.

“It’s so important to increase participation of women and minority groups in STEMM so I initiated the establishment of SAGE – Science in Australia Gender Equity, to support and retain women researchers.”

Professor Joshi has also recently been elected Vice-President of the International Mathematical Union at its General Assembly in Sao Paulo, Brazil.

Shedding light on some of the world’s most difficult problems all while being an ambassador for women and minority groups, Nalini Joshi is certainly making an impact.