The analysis group in Helsinki consists of nine senior researchers, three post docs and 20 graduate students. This does not include people in mathematical physics and applied analysis whose research is often closely connected with strong co-operation. Large part of the research of the group has a historical back-ground in the famous Finnish complex analysis school of Lindelöf, Nevanlinna, Ahlfors and others, and the later foundational work on quasiconformal and quasiregular mappings of Lehto in two dimensions and of Martio, Rickman and Väisälä in higher dimensions. The contacts to these traditions are still clear and visible in much of the present day work although the research has diversified to many directions with many new connections and applications. This is clear from the following brief description of some recent research topics and results:
Astala has solved together with Päivärinta a well-known problem of Calderón applying quasiconformal mappings to partial differential equations and tomography. Martio together with Gutlyanskii has introduced a new tool of conformal differentiability of a quasiconformal mapping to study boundary behaviour of conformal maps. Holopainen has studied quasiregular mappings on Riemannian manifolds and Carnot groups. Martio and Holopainen have developed nonlinear potential theory in metric spaces. Saksman works on harmonic, stochastic and geometric analysis. In applied side he has studied computational MCMC methods and inverse problems. Tukia is doing fundamental work on Kleinian groups and Teichmüller spaces. Seppälä works on computational conformal geometry. Hurri-Syrjänen has studied Sobolev, Besov and other related function spaces. Mattila has worked on geometric measure theory and its applications to complex and harmonic analysis. Taskinen is working on functional analysis and nonlinear parabolic PDE's. Tylli's research concentrates on geometry of Banach spaces and operator theory. The post docs Harjulehto, Hästö and Koskenoja are working on variable exponent spaces.
The group has strong international and national contacts. Most of the papers are joint papers with foreign researchers. Visits to and from the group are frequent at all levels; senior researchers, post docs and graduate students. The group has a well established international reputation as evidenced, for example, by many invited talks at high level conferences (three members of the group have been invited speakers at ICM, two at ECM). The group is one of the main ingredients in organizing Nevanlinna colloquia which have become one of the most respected regularly held mathematical analysis conferences. On the national level co-operation with Jyväskylä is very intense; majority of the group belongs to the joint Center of Excellence with the Jyväskylä analysis group. There are also good co-operation with the Technical University of Helsinki and Universities of Joensuu, Oulu and Turku. Co-operation with other research groups inside the department is active, in particular with mathematical physics and applied analysis.