The research group is composed of active researchers with diverse backgrounds in mathematical analysis and mathematical modelling with applications to biology and ecology. Our three main domains of interest are mathematical ecology, complex analysis and partial differential equations.
Mathematical ecology is an interdisciplinary research area where advanced mathematical and computational tools are used to study issues in ecology and evolution. At Umeå, research on these topics involved not only members of the Analysis and Mathematical Modelling group from the department of Mathematics, but also researchers from the Department of Ecology and Environmental Science (EMG) at the interdisciplinary Icelab
Partial Differential Equations
Our research concerns mainly nonlinear Partial Differential Equations (PDE) including e.g. the p-Laplace equation - a natural nonlinear generalization of the well known Laplaces equation. Such PDEs have connections to, e.g., minimization problems, nonlinear elasticity theory, fluid dynamics, stochastic games and image processing.
Some typical problems we study are the existence and uniqueness of viscosity solutions, growth rates of positive solutions vanishing on a portion of a boundary, strong maximum, minimum and comparison principles, Phragmen-Lindelöf theorems, i.e. estimates of growth of solutions far away from a boundary, construction of explicit solutions, system of PDEs with obstacles with applications to optimization problems such as e.g. managing a chain of hydro power plants.
Besides creative, elaborate constructions and lots of geometric thinking, our research on differential equations involves real analysis, integration theory, measure theory, regularity theory, probability theory and other interesting things.
We do research mainly in several complex variables and our research is focused upon pluripotential theory and certain aspects of analytic function theory. Pluripotential theory is a nonlinear analog of the classical potential theory for subharmonic functions. The main objects of study are plurisubharmonic functions and the complex Monge-Ampere operator (ddcu)n – a generalization of Laplace's operator. In the theory of analytic functions we are mainly interested in ideals and spectrum for algebras of analytic functions, in particular Gleason's problem. In connection with this, we are also interested in domains of existence and envelopes of holomorphy for classes of analytic functions.