Wiśniewska is intrigued by this thrill and the sense of exploring new territory: “There are barely any geographical locations left that no human has seen before. In mathematics, on the other hand, we keep discovering new places. The only border is our own mind,” she says. “This sense of exploration and discovery in mathematics never ceases to amaze me.”
In Wiśniewska’s field of specialism, a sense of adventure is a great advantage. Symplectic geometry is a very young sub-discipline that emerged only in the mid-1980s. Its foundations have remained the subject of intense debate even to this decade. It has its beginnings in the classical physics of the 19th century: in principle, it investigates the position and movement of particles in space and how they relate to energy.
Humans encounter examples of those systems in their everyday life. Even though these systems can be described by relatively simple mathematical equations, they may be too complex to be solved explicitly with a precise formula.
The task is nontrivial even in simple, bounded spaces such as spheres and tori, but becomes even more challenging in non-bounded spaces like cosmos. The field of symplectic geometry is full of unsolved mathematical questions and puzzles to this day.
Closed orbits and open paths
When a particle moves through space and returns to its original position, mathematicians talk about a closed orbit. The starting position is called the fixed point. This applies to planets that orbit stars and – as in Wisniewska’s research – satellites that orbit planets.
Mathematicians can use such fixed points and closed orbits to determine the properties of spaces and forms, as the example of buns and doughnuts illustrates. Wisniewska wants to find out where and how a satellite can be placed in space to ensure that it remains in a closed orbit and does not collide with a planet or drift away into infinite space.
Finding the right orbit is not easy. The shape of the set with constant energy can provide important inferences as to whether or not there is a closed orbit.
Wiśniewska grew up in Poland and completed her degree at the University of Warsaw. During an Erasmus exchange in the Netherlands, she met Federica Pasquotto, who became her doctoral supervisor at the University of Amsterdam in 2017. For the past year, Wiśniewska has been a postdoctoral researcher at ETH in Will Merry’s group.
She enjoys the working environment in the ETH main building and the open exchange within the group. This is important to her: “Mathematics thrives on the exchange of views and on mutual encouragement to try your own ideas.” How her career will progress is still open.
But she has a message for schoolgirls who are contemplating a degree in mathematics: “Give it a try. You can do it. There are no borders in mathematics.”