Susanna Heikkilä uses knitting to illustrate the essence of mathematical problems
Mathematics includes a domain called topology, which examines spatial forms and surfaces as well as interrelationships between differently shaped mathematical objects. More specifically, Susanna Heikkilä’s particular field of mathematics is called quasiconformal geometry, a domain of geometrical analysis.
At her work at the Ģֱ, Heikkilä is doing basic research that is connected, for example, to inversion problems drawing on the methods of partial differential equations, harmonic analysis and geometry. Inversion problems have many practical applications, such as in medical imaging (X-ray and ultrasound imaging) and soil survey.
Surface modelling by knitting
For her presentation section at her doctoral defence, Heikkilä considered how she could illustrate the concepts and connections of her dissertation to her relatives as concretely and comprehensibly as possible.
“I like knitting and noticed that a knitted plane can be twisted and stretched,” Heikkilä says.
“I realised that by means of knitworks I can easily model topological forms and move from a flat surface to a ball-like surface.”
Heikkilä has a plane-like piece of knitwork on her desk. She has also a ball with equator markings like on a globe. She places the piece of knitwork on the ball and fixes the corners at the equator. It soon becomes clear that some gaps remain between the knitted pieces and the ball.
“When I stretch a knitted piece, the gaps on the ball are covered,” Heikkilä says. “This stretching depicts quasiconformal geometry and my research in simpler terms.”
Solving a 40-year-old problem
In her doctoral dissertation, Heikkilä solved a mathematical problem that had bothered experts of this field for more than forty years. The problem dealt with quasiregular representations and how four-dimensional shapes can be presented by means of four-dimensional Euclidian space geometry.
For mathematicians, a four-dimensional world is not any stranger than living in a three-dimensional world,” Heikkilä says.
Quasiregular representations are interspatial descriptions, which alter angles and shapes to a limited extent. These representations are directional and provide a way to move from one space to another.
In 1981, the Russian-French mathematician Misha Gromov considered whether there is any such obstacle for the existence of quasiregular representation that is not related to the gaps measured by the target side fundamental group.
“The target side refers to the space entered,” Heikkilä says. “The spatial fundamental group measures gaps around which a rope held from both ends gets stuck. For example, a rope curled round an infinitely tall light pylon gets stuck unless either end is released.”
In addition, Gromov also asked that if the target side of a representation is simply connected, is the existence of the representation guaranteed? The question remained open for nearly 40 years. Not until in 2019, when mathematician Eden Prywes was able to offer a four-dimensional counter example to this question, which brought the final solution a bit closer.
My dissertation supplements the answer to Gromov’s question, because it can be used in classifying four-dimensional simply connected objects, for which there is a quasiregular representation of Euclidian space,” Heikkilä states.
The solution concerned is part of basic research in mathematics, and it can help other mathematicians achieve new research results in the future. The article related to the doctoral dissertation was approved for publication in the highly regarded journal . Heikkilä had been struggling with the problem since her master’s thesis.
It felt really great to solve a problem I had spent several years on,” Heikkilä says.“
"Even if I achieved nothing else in my career after this, I have accomplished something significant at a relatively young age. On the other hand, such a great achievement in the academic world creates some extra pressures for my further research.”
Topology was the inspiration to continue with mathematics
When still at upper secondary school, Heikkilä did not know what she would like to study after graduation. Even though she was otherwise good at mathematics, she made a lot of miscalculations. She ended up applying for a University School of Business and Economics, but when preparing for the entrance examination she realised this field was not for her. With encouragement from her former teacher, she moved to plan B and applied for a study place in mathematics.
She soon realised that university mathematics is totally different from mathematics at upper secondary school, and it suited her much better.
I was still uncertain of the field of studies during my first year, however,” Heikkilä remembers. “
During my second-year studies, I took a topology course, which convinced me to continue with mathematics. When I found an interesting topic for myself as well as the right kind of study methods, it was easy to go on with the studies.”
Conformal and quasiconformal maps as aids in facial recognition
Heikkilä completed her doctoral degree at the University of Helsinki in 2024. In January 2025, she started as a postdoctoral researcher at the Ģֱ, in Professor Kai Rajala’s research team. Rajala was also one of the two pre-examiners of Heikkilä’s dissertation.
Rajala and Heikkilä are studying quasiconformal analysis in a project funded by the Research Council of Finland. The purpose is to apply modern methods of mathematical analysis to the investigation of shapes of unsmooth and fractal spaces.
“Our abstract research targets are closely related to topical and concrete problems, such as facial recognition,” says Rajala.
“We are looking for conformal and quasiconformal coordinates, or maps, which distort the shapes of small objects as little as possible.”
The research targets include both classic representation problems that are over a century old, along with modern applications of map representation theory, such as the classification of abstract algebraic structures and dynamic systems.
Pencils, chalk and felt-tip markers are a mathematician’s tools
Nowadays, one could think that blackboards and whiteboards belong to the past. However, in contemplative work and sharing different ideas such boards are still irreplaceable. You can see at a glance everything written, and different boards are used in testing and exchanging various ideas among colleagues. In addition, Heikkilä does all her initial work with paper and pencil.
“I do fairly little research on a computer,” Heikkilä says.
I spend my workdays reading other researchers’ publications and reflecting on mathematical solutions with a pile of sketch paper. Using blackboards and whiteboards, we consider problems together with colleagues.”
Weekly, there are many research seminars at the Department of Mathematics and Statistics which Heikkilä follows. Seminars offer new research knowledge for broadening your views and thinking. The department’s close-knit and collaborative atmosphere has impressed her.
“I have had a good time here in Jyväskylä and integrated well to the Department of Mathematics and Statistics,” Heikkilä says. “Other researchers have helped me, and it has been nice to exchange thoughts with them. I am looking forward to the future and what it will bring.”
