Department of Physics and Mathematics

In photonics one investigates light, its properties, and its utilization in various applications in, e.g., ICT and medical sciences. Photonics research and education at UEF is under the Center for Photonics Research. See list of research topics here. 

Complex Analysis, or Complex Function Theory, is a field of mathematics which studies analytic or meromorphic functions, integration and mappings in the complex plane or its subsets.

Value Distribution Theory

Nevanlinna theory (or value distribution theory) deals with the growth and value distribution of meromorphic functions. One of its central results is the second main theorem, which is a deep generalization and quantification of Picard's theorem. The basic theoretical structure analogous to Nevanlinna theory can be found in many areas of mathematics such as p-adic function theory, minimal surfaces and even Brownian motion. According to the work of Osgood and Vojta the second main theorem of Nevanlinna theory corresponds to the ABC conjecture in number theory, which in turn implies asymptotic version of Fermat's Last Theorem. Another interesting analogue is the so-called Tropical Nevanlinna theory discovered by Halburd and Southall, which deals with piecewise linear real functions over a max-plus semiring. Applications of Nevanlinna theory can be found mostly in other branches of mathematics, such as complex oscillation differential and functional equations, or in areas adjacent to mathematics, such as mathematical physics.

Operator Theory and Function Spaces

Research on operator theory concentrates on concrete operators such as the Bergman projection, Toeplitz, Hilbert, integral and composition operators acting on spaces of analytic functions in the unit disc employing harmonic and functional analysis. In function spaces the main focus lies on small Bergman spaces whose harmonic analysis is somewhat similar to that of the Hardy spaces and is therefore challenging compared to the case of standard Bergman spaces.

Differential Equations

The long development of theory of linear differential equations in the complex domain has created an extensive network of international collaboration. One of the starting points has been the study of growth of solutions in case of the complex plane, from which researchers have proceeded to consider similar problems in the unit disc. The Joensuu research group has been particularly strong in applying the theory of analytic function spaces in differential equations.

Another central theme of recent research activity has been the oscillation theory. For example, in case of the unit disc the study of oscillation of solutions is an interesting combination of value distribution theory of meromorphic functions, function spaces, univalent functions, interpolation and non-Euclidean geometry. One of the main objectives is to describe the geometric zero distribution of solutions.

After some quiet years the research on complex differential equations in the case of plane has started a new rise. The subjects of research are now special solutions in terms of canonical products and contour integrals, and the oscillation of solutions in the case when the coefficient functions are exponential polynomials, to name a few. Moreover, cases in which the coefficient functions are special functions have recently attracted interest.  

You can find the contact information of the group from the Complex Analysis research group page in UEF Connect.

The research on partial differential equations can be divided into two subtopics: overdetermined problems and their numerical solution, and nonlinear potential theory.

Overdetermined problems occur both in pure and applied mathematics. The framework for analyzing such systems, the formal theory of partial differential equations, is based on considering a given equation as a submanifold in a suitable jet space. We have shown that the central concept of the theory, the involutive system, is also useful in the numerical solution of partial differential equations. This research is done in collaboration with Bijan Mohammadi (Université de Montpellier).

The algebraic tools which are also useful in the formal theory has also lead to the analysis of some constrained systems in multibody dynamics. Here the focus is on the analysis and characterization of possible singularities of the configuration space of the multibody system. This topic is studied in collaboration with Samuli Piipponen (University of Tampere) and Andreas Müller (Johannes Kepler Universität Linz).

The research on nonlinear potential theory concentrates on the regularity theory of solutions and supersolutions of nonlinear elliptic problems in metric spaces and in variable exponent spaces. In particular, the potential theoretic problems related to the fine topology are studied. The research related to the variable exponent is a part of the national research network Finnish variable exponent Sobolev spaces research group. The research on metric spaces is based on the co-operation with Anders and Jana Björn (Linköping University) and Tomasz Adamowicz (Institute of Mathematics, Polish Academy of Sciences).

Research on teaching and learning of physics and mathematics is based on the traditions of Physics Education Research (PER) and Mathematics Education Research. The research produces detailed knowledge about the factors influencing student learning. The results are used as a base in developing novel solutions for teacher education in mathematics and physics and for school teaching as well.

The most salient themes related to teacher education are teacher knowledge of mathematics and physics teachers and students’ mathematical-related orientations. Mathematics in engineering education is studied in cooperation with Karelia University of Applied Sciences. The projects belonging to the LUMA FINLAND programme focus on two themes: 1) the integration of technology education in physics and chemistry teaching at school, and 2) reducing the effects of sex segregation in students’ education and career choices in science and engineering fields.

Contact person:

• physics or mathematics education research or doctoral education: Asikainen Mervi, Senior Lecturer
• issues related to LUMA: Saana Kinnunen, coordinator

You can find the contact information of Physics, Mathematics and Technology Education Research group from UEF Connect.