29.05.2018

Geometry and Analysis

The Geometry and Analysis group studies a wide range of topics, including Analysis on Metric Spaces, Geometric Measure Theory, Partial Differential Equations, Calculus of Variations, Geometric Analysis and Inverse Problems.

The Group is part of the  Finnish Centre of Excellence in Analysis and Dynamics Research for the years 2014-2019 together with the analysis and mathematical physics groups at the University of Helsinki and the University of Oulu. The inverse problem research group is a member of the Finnish Centre of Excellence in Inverse Problems, which is internationally recognized as the world's leading unit in the field.

Many group members are funded by the European Research Council (ERC) and Academy of Finland grants.

Members

Some former members

Activities

Upcoming events

Some research interests

Geometric Mapping Theory

The basic theme of our research is the study of the analytic, geometric and topological properties of mappings under various analytic assumptions. This often leads to the need to understand the geometries of the underlying spaces. We study modern areas, most of which have their origins in geometric function theory. Mapping problems naturally appear whenever one looks for parametrizations, minimizers, or solutions to analytic equations. Besides analysis, our research is directly connected to several different areas of mathematics, such as differential geometry, partial differential equations, geometric topology, and geometric group theory.

Inverse Problems

Inverse problems research concentrates on the mathematical theory and practical implementation of indirect measurements. Applications are found in numerous research fields involving scientific, medical or industrial imaging; familiar examples include X-ray computed tomography and ultrasound imaging.

The inverse problems research group in Jyväskylä focuses on fundamental theoretical aspects of inverse problems such as the Calderón problem in electrical imaging and travel time tomography in seismic imaging. Inverse problems have a rich mathematical theory employing modern methods in partial differential equations, harmonic analysis, and differential geometry.

Nonlinear PDEs

Partial differential equations have a great variety of applications to mechanics, electrostatics, quantum mechanics and many other fields of physics as well as to finance. In the linear theory, solutions obey the principle of superposition and they often have representation formulas. However, it is sometimes said that the great discovery of the 19th century was that the equations of nature are linear whereas the great discovery of the 20th century was that they are not. Nonlinear PDEs appear for example in stochastic game theory, non-Newtonian fluids, glaceology, rheology, nonlinear elasticity, flow through a porous medium, and image processing. Since superposition is not available, methods needed to study nonlinear equations are quite different from those of the linear theory.

Non-smooth Geometry

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.

Publications

Publications of the group can be found on the arXiv preprint server and the TUTKA database. Here is a list of recent publications of the group members.