Understanding dynamical systems Mathematical billiards provides a playground for researchers to investigate dynamical systems and to look at how they evolve in time. Professor Vadim Kaloshin and his colleagues in the SPERIG project are tackling the Birkhoff conjecture, an open question in the mathematical billiards field, which could open up new insights into the behaviour of dynamical systems. A
dynamical system by nature changes over time, and many mathematicians have devoted their attention to investigating the underlying laws behind this evolution, from Newton’s observations of celestial mechanics to Poincaré’s analysis of autonomous systems. In the case of a billiard table, the evolution of the system can be described if the angle at which the ball hits the boundary of the table is known. “If you want to see where the ball will land after hitting the boundary of the table, you need to record an angle. The angle of reflection is equal to the angle of incidence,” explains Vadim Kaloshin, Professor of Mathematics at the Institute of Science and Technology Austria (ISTA). A billiard table is of course conventionally rectangular, but in his research, Professor Kaloshin also explores the behaviour of balls or particles in other shapes. “We might think of a billiard table that is circular, for 58
example, not rectangular. The common law is that the angle of reflection is equal to the angle of incidence. This then becomes an interesting mathematical problem,” he outlines.
SPERIG This topic is a central part of Professor Kaloshin’s work as Principal Investigator of the ERC-backed SPERIG project, an initiative based at ISTA. Together with colleagues
Diagram based on the ‘Bunimovich stadium’.
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in the research group he leads at ISTA, Professor Kaloshin has been studying two related questions in the dynamical systems field. “In the SPERIG project, we have been looking at a mathematical billiards problem and a problem around geodesic flows. These two problems are similar in some ways; they can be thought of as siblings to an extent, but they are nevertheless different from each other,” he outlines. The mathematical billiards problem that researchers are addressing in the project centres around the Birkhoff conjecture, which in extremely general terms states the circumstances under which the boundary of a billiard table must be an ellipse. “The dream is to prove the global Birkhoff conjecture,” says Ilya Koval, a PhD student in Professor Kaloshin’s group. The table in mathematical billiards is considered simply as a boundary, and the ball as something that moves without friction, from which researchers then seek to derive more general principles about the way that a ball moves across the surface. Mathematicians study billiards as a collision map, moving from one collision to another, essentially looking to describe the map. “Dynamical systems is an area of maths where classes of maps are analysed, and where we seek to discover various properties of those maps – for example, whether the orbits of the balls are stable or not, or whether there are some invariant sets or periodic orbits,” explains Professor
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Kaloshin. The movement dynamics seen on a billiard table can also be found in other systems, for example, in the way that sound waves move, a topic that Professor Kaloshin is exploring in the project. “I’m trying to prove that if you deform a drum, then you also deform a sound,” he says. This specific question has a long history, dating back to the work of Hermann Weyl,
a highly influential figure in the history of mathematics. In its more modern form, the question was popularised by the PolishAmerican mathematician Mark Kac in a 1966 research paper; now Professor Kaloshin and his colleagues are tackling it anew. “The goal is to reach a point where we can say that if you try to deform a drum, of any type, you will deform a sound,” he outlines. This research is
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