Quasi-conformal and related mappings form the largest class of maps that can be studied by analytic methods. Accordingly, their theory and their applications lie at the intersection of geometry and analysis and have connections to many other areas of mathematics and physics. While fundamental questions at the foundation of the theory of quasi-conformal maps remain open, recent advances in analysis on metric spaces have enlarged the range of applicability of these maps. Their theory may now lead to solutions of previously inaccessible problems in other fields such as geometric group theory. The purpose of this project is to explore current trends in the area. Specific topics of research include quasi-symmetric uniformization, dynamics on fractal spheres, the quasi-conformal Jacobian problem, quasi-regular maps and elliptic manifolds.
The roots of this subject can be traced back to Gauss's work on cartography and surface geometry in the first half of the 19th century. He coined the phrase ``conformal map" and derived equations that govern the theory of planar quasi-conformal mappings. The full importance of this theory was only realized about a century later. By now quasi- conformal mappings have developed into one of the major tools in contemporary Geometric Function Theory.
Intellectual merit: A core topic of the current project was the investigation of self-similar fractals mostly related to their quasiconformal geometry. Fractal and self-similar features can be seen in many natural phenomena such as coast- and fault lines, snowflakes and crystal growth, electric discharge or plant growth patterns. This project contributed to the development of basic mathematical methods and tools that are necessary for a deeper understanding of such structures. A key concept for this investigation was the notion of a quasiconformal map. Such a map can be thought of as a deformation of a structure (such as a fractal) that distorts its small-scale geometry by a controlled amount. One is interested in the overall global effect of such a deformation. In the project such questions related to quasiconformal geometry where explored for a particularly important class of fractals---Sierpinski carpets and fractal spheres. In the mathematical context these fractals often arise from the dynamics of groups or the dynamics of maps under iteration, and often a better understanding of their quasiconformal geometry is of crucial relevance. Broader impact: This project enlarged our understanding in a relevant subject of mathematical research. An important part of the activity was the involvement of young researchers. It provided them with the mathematical expertise necessary for independent investigations of fractal geometry, and will help to maintain a scientific community which provides the necessary mathematical knowledge for progress in science and engineering.
