Objective
The interplay between potential theory, value distribution theory, complex dynamics and geometric function theory has been decisive for several deep results recently. The overall objective is
(1) to conduct active, co-ordinated search for such interplay ideas;
(2) Evaluating consequences of these ideas aims;
(3) to extend the present knowledge in the above and related Fields such as;
(4) complex differential equations and complex function spaces. Practical consequences will be approached by;
(5) investigating the intersection of value distribution theory and applied mathematics and by (6) considering applications to control theory and optimisation;
(7) All teams will be active in doctoral and post-doctoral training, including opening of internal workshops for outsiders.
Scientific research to be done: In potential theory, the standard non-linear potential theory (corresponding to quasi-regular mappings) will be extended to the non-linear pluripotential theory (corresponding to functions of several complex variables). Classical subharmonic functions will also be studied, in relation to the value distribution theory. In particular, the celebrated star function idea will be extended by providing a general scheme of constructing operators preserving subharmonicity. Studies in fine potential theory aim to attack several difficult problems in analysis.
In value distribution theory, a key idea is to study locations of a-points of meromorphic functions, similarly as to the classical theory describing the quantities of a-points. This will be approached by the method of gamma-lines and by studying Fourier coefficients of logarithms of meromorphic functions. Applications of gamma-lines into physics and environmental problems will also be studied. To extend classical plane results to the unit disk, order considerations of meromorphic functions of slow growth will be performed, as well as studies of the link between normal families and value distribution theory.
In complex dynamics, the dynamics of transcendental functions will be investigated by looking several questions previously well understood for rational functons. A systematic investigation will be made on wandering and Baker domains, including the interplay between complex dynamics and complex differential equations. In the latter field, applying the location studies above will perform considerations on locations of a-points of solutions. Normal family arguments and complex function space ideas will be applied to complex differential equations.
In geometric function theory, the present knowledge of quasi conformal and quasi regular mappings will be extended to a theory of multi-valued quasi regular mappings. Local structure considerations needed here will be applied to finding extreme metrics and moduli in non-orientable Riemannian manifolds. Moreover, similar extremal problem ideas in univalent functions will be applied to approximation theory.
Complex function spaces and operator theory will be studied to build up a strong bridge to value distribution theory in addition to the relation with the star function ideas. Extreme problem ideas similar as to described above may be combined with control theoretic and optimisation methods to obtain new important extremal problem results for linear functionals.
Dissemination of research results: All research results will be published in high-level refereed journals and, if relevant, as monographs through international publishers. Before published, the results will be made accessible for the outside mathematical community by electronic communication.
Call for proposal
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80101 Joensuu
Finland