Geometric Mapping Theory
Overview
Research group description
The basic theme of our research is the study of the analytic, geometric and topological properties of mappings. Mapping problems naturally appear whenever one looks for parametrizations, minimizers, or solutions to analytic equations. Already in their definitions, but even more so in their study, a rich interaction between geometry and analysis arises.
Classically, the field has origins in geometric function theory. However, the field has connections and applications in a wide variety of disciplines: differential geometry, partial differential equations, geometric topology and geometric group theory.
Research Interests
Analysis on metric spaces
(P. Koskela, A. Koski, J. Onninen, K. Rajala, S. Eriksson-Bique and V. Tengvall)
The purpose of this modern area is to develop analysis in non-smooth, or even fractal spaces. New applications to problems where smoothness is not present appear all the time. We study weakly differentiable mappings in singular spaces: Sobolev and BV-functions, Lipschitz and quasiconformal mappings. The Poincaré and isoperimetric inequalities as well as heat kernel and other analytic estimates are studied both independently and as tools to find out the geometric properties of the underlying spaces. The theory of analysis in metric spaces has shown in particular that surprisingly many deep results in analysis only have little dependence on the structure of the underlying space.
Geometry of domains
(P. Koskela, A. Koski, J. Onninen, K. Rajala, T. Rajala and V. Tengvall)
Euclidean domains that admit an extension, say for each p-integrable Sobolev function with p-integrable first order derivatives to the entire Euclidean space can be used as a substitute for the entire space in many questions. Such domains also form natural examples for analysis on metric spaces. Sometimes already the validity of a Poincare type inequality is sufficient. We have studied geometric criteria for extendability of Sobolev functions or for the validity of such inequalities. Many challenging problems remain. Our research on the Hardy inequalities is also related to the geometry of domains. We study in particular the connections between the validity of these important inequalities and the size and geometry of the boundary of the domain.
Sobolev mappings and non-linear analysis
(P. Koskela, A. Koski, J. Onninen, K. Rajala, and V. Tengvall)
Minimizers of suitable energy integrals modeling the physical properties of, say, elastic materials, are sense-preserving Sobolev mappings with some analytic properties. These properties are often familiar in quasiconformal analysis, although weaker. The simplest example is that of a harmonic map. We study the properties of such mappings under minimal regularity assumptions, motivated by the connections to both non-linear elasticity theory and quasiconformal analysis. We try to find the minimal assumptions guaranteeing continuity or partial continuity, invertibility, preservance of sets of measure zero, and regularity of the mappings as well as their inverses; properties which have natural physical interpretations. Similarly, we try to find sharp extensions of the basic results in quasiconformal analysis and geometric function theory.
Quasiconformal analysis
(P. Koskela, A. Koski, J. Onninen, S. Eriksson-Bique, K. Rajala and V. Tengvall)
Quasiconformal and quasiregular mappings extend and generalize conformal maps and complex analytic functions, respectively. In particular, the theory developed for them over the past fifty years extends geometric function theory to higher dimensions. Although the theory is already extensive, there are several fundamental problems that remain unanswered, such as the optimal regularity of mappings, the rigidity of higher dimensional mappings compared to the two-dimensional, boundary behavior, as well as the topological properties. More recently, the theory has been extended to cover mappings between singular spaces, with succesful applications to several problems, for instance in geometric group theory and in the classification of metric spaces.
Measure and dimension theory in metric spaces
(J. Lehrbäck, K. Rajala, T. Rajala and A. Vähäkangas)
Along with Euclidean spaces, mathematical analysis is focusing more and more into general metric spaces. As these abstract spaces often have fractal features, it is very important to develop the machinery of geometric measure theory in these spaces in order to get better understanding of the local structures. Many of the research themes discussed above are related to and make sense in quite general metric spaces.
