What are your research topics?
I study how mathematical objects of varying shapes and dimensions react to each other. For instance, can bodies be reshaped to exchange forms without breaking their internal structure, when can an abstract mathematical object be realised in a specific space, or which preconditions make it possible to wrap an object onto an ideal surface?
More specifically, my field focuses on scale-independent non-smooth geometric phenomena. Within mathematics, this field is known as quasiconformal geometry, which is a part of the discipline of geometric analysis.
Where and how does the topic of your research have an impact?
Quasiconformal geometry constitutes basic mathematical research. Its results are utilised within mathematics in other fields of basic mathematical research, including geometric group theory and geometric topology.
Since research in the field focuses, broadly speaking, on all geometric phenomena that are related to shapes and their preservation in geometric variants, it is easy to identify points of contact with physics, computer science and other fields of science neighbouring mathematics. Such links include questions of geometry relating to networks. At the same time, questions related to the geometric properties of materials connect, through inverse problems, quasiconformal geometry with, for example, medical imaging.
What is particularly inspiring in your field right now?
Quasiconformal geometry really comes into its own in circumstances where the object studied shares the same dimensions with the surrounding space. Very recently, our research group found how methods of quasiconformal geometry can be applied to studying mappings of higher-dimensional spaces, also known as embedding questions.
This observation has created new and surprising connections between quasiconformal geometry and mathematical fields such as the theory of several complex variables, calibrated geometry and the theory of minimal surfaces.
Pekka Pankka is the professor of mathematics at the Faculty of Science.