Future Proof — Angkana Rüland

Angkana Rüland has been a Max-Planck research group leader of the group Rigidity and Flexibility in PDEs at the MPI MiS since October 2017. In joint work with her group she has investigated inverse problems, nonlocal equations, problems from the calculus of variations and free boundary value problems. Before starting at the institute, she studied Mathematics in Bonn and Leipzig and had been a postdoctoral research associate at the University of Oxford. In April 2020 she will become a professor of applied mathematics at the Ruprecht-Karls-University in Heidelberg.

Unique continuation and inverse problems

Inverse problems are ubiquitous in nature: Animals like bats navigate with sonar, medical applications like X-ray tomography allow for non-invasive measurements and diagnoses and spectroscopy methods allow to indirectly determine the composition of chemical substances. In all of these examples one is interested in reconstructing information in various settings from physics, engineering or medicine for which only indirect, non-invasive measurements are available. The theory of inverse problems for which unique continuation results yield important tools allows one to deduce such results in a mathematically precise way.

Free boundary value problems and their regularity properties

Free boundary value problems arise in many processes in the sciences and our daily lives. A prototypical example for instance includes the melting of ice in water, the so-called Stefan Problem. While there are evolution equations for the ice and the water, the interface of the ice and the water is not explicitly given in the description of this problem. It is part of the problem to determine, to predict and to analyze its evolution. This makes these problems very “nonlinear” and poses interesting mathematical challenges in particular in terms of the regularity theory of the solutions and interfaces.

Problems from the calculus of variations in particular the dichotomy between rigidity and flexibility in the modelling of shape-memory alloys

Shape-memory alloys are materials with a thermodynamically very interesting behaviour making them promising materials for a number of industrial applications. Just as in the melting of ice to water, these materials also undergo a phase transformation, however from one solid state with very high symmetry at high temperatures to another solid state with less symmetry at low temperatures. This loss of symmetry implies that the materials have different variants of their low temperature state and are thus very flexible. When heating up the material again, it loses this flexibility and has to recover its original shape – it has “a memory”. Mathematically, the presence of multiple variants of the low energy states provides these materials with a rich energy landscape leading to a variety of different microstructures. It has been one of the objectives of the research group Rigidity and Flexibility in PDEs to study their rigidity and flexibility properties.