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.
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.