The challenges of mathematics equate to the most exciting new frontiers of science.
Imagine taking a time-lapse photograph of a clear sky at night. The photograph will be filled with circular arcs of light that reflect the motion of the stars in the sky as the Earth rotates around its axis. These paths have been the subject of human wonder since the time of ancient civilisations, and our precise mathematical knowledge of the positions of stars has enabled us to navigate our way home from long journeys across vast oceans.
Now imagine you are on a distant planet, whose rotation is not so regular or predictable. It may have a cycle of rotation that gets shorter or longer, according to which of its suns is nearby. How do we build mathematical tools to navigate our way home on such a planet? This question might seem like the start of a science-fiction novel. But actually, the same questions arise in many settings, including our day-to-day life on Earth.
If you have ever waited for a bus, you will know about one of these settings. The precise arrival times for buses (at least in my home town) are unpredictable. The published timetable might suggest that the bus you are waiting for is arriving in four minutes, but you have no way of knowing whether it has already arrived and departed before you got to the bus stop. In most cases, it will be delayed, and you will not know how long it has been delayed. In my experience, often the next bus catches up with it and both buses arrive simultaneously at the stop where everyone is waiting.
In Cuernavaca, Mexico, a private bus system has evolved to overcome some of these problems. A driver pays a small fee to an observer at each stop for information about when a previous bus departed on the same route. If the departure time was recent, the driver waits. If the departure time was some time ago, the driver departs. They adjust the waiting times and speeds so as not to lag too far behind or get too close to the next bus. It is an example of a system designed to maximise passenger numbers on each bus while minimising waiting time between buses.
The beauty of mathematics is that the description of the Cuernavaca bus arrival times also works in other situations where there is attraction and repulsion between objects. Instead of buses, think about subatomic particles interacting with each other in a particle collider far under the ground in Switzerland. Instead of particles, think about large prime numbers and how they are spaced apart from each other on the number line.
A prime number is a positive integer that is divisible only by itself and by the number one – some small prime numbers are 2, 3, 5, 7, 11, 13, …, while some larger ones are 88,969, 200,023. Large prime numbers form the basis of the RSA algorithm, which is a public-key cryptographic system widely used to ensure secure data transmission. There is no predictive algorithm for prime numbers. Pairs of successive prime numbers differ by as little as 1 or as much as 1,113,106. The search for larger and larger primes continues. As of December 2020, the largest known prime number is 282,589,933 _ 1, a number which has 24,862,048 digits (in base 10).
Prime numbers are related to the zeroes of a function called the Riemann zeta function. There is a famous unsolved problem in mathematics called the Riemann hypothesis, which asserts that the zeroes of this zeta function must lie on a certain vertical line in the complex plane. (This is the subject of a Millennium Prize Problem, whose verified proof will net you a $1 million prize.) So the study of large zeroes of the zeta function is a very active area in mathematics. The spacing of these large zeroes turns out to follow a law that also describes how sub-atomic particles are repelled in a scattering experiment, and how the statistics of the bus system behave in Cuernavaca, Mexico.
The statistics of such spacings are worked out through mathematical models that I study. But there is a frontier that has not yet been crossed. In the settings I described, time changes continuously. What would happen if the clock available to us changed time in discontinuous steps of variable length?
In our time-lapse photographs of stars in the sky, time changed continuously. If we took a photograph once per hour, instead of leaving the camera aperture open for a long length of time, we would still get the same information about the stars (we can interpolate smoothly between the snaps to describe the paths taken by the stars, because we know how the Earth rotates). But the problem is harder if our camera is only allowed to take snaps at times that are variable, sometimes as short as a minute apart or as long as two hours apart. The problem may not be solvable on the alien planet whose rotation is unpredictable.
I work on problems arising from physics where the time-stamps on the clock are not regularly spaced. For example, they change multiplicatively (ie, one timestamp t changes to qt, where q is a nonzero number not equal to unity), or they change through a more complicated pattern given by certain functions called elliptic functions.
I am thrilled by the prospect of discovery, the possibility that the mathematics I develop will lead to new connections and new models that will describe how the most elusive structures in the world change with time.
Professor Nalini Joshi
Professor Nalini Joshi, AO, is Chair of Applied Mathematics at the University of Sydney and a Payne-Scott Professor. She is a world leader in the theory and applications of differential equations, contributing mathematical results that have impact in fields as diverse as particle physics, quantum mechanics, large prime-number distributions, and wireless communications. Her distinguished research record has led to numerous awards.