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Some novel Nevanlinna type results in different branches of mathematics (Nevanlinna)
Date du début: 19 août 2013, Date de fin: 18 août 2015 PROJET  TERMINÉ 

"Fundamental results in Nevanlinna value distribution theory (one of the greatest achievements of the century, in words of Herman Weyl) and the theory of covering surfaces by Ahlfors (who was among the very first Fields medalists) complemented the list of results of general nature in the complex analysis (i.e. those results that relate to the most important and largest classes of analytic or meromorphic functions).Since 1970s several new results of similar general nature were obtained. Concerning earlier publications see the book [21], 2002, where nearly 10 preceding results related to arbitrary meromorphic functions are presented; concerning recent publications see papers [31] and [34] involving more than 15 results related to the very large classes of plane curves, real functions, algebraic functions as well as to solutions of large classes of equations.Most of these results are closely connected with the Nevanlinna and Ahlfors theories and, in fact, transfer ideas and methods of these theories (complex analysis) into real analysis (where we study highly applicable level sets of real functions), differential geometry (where we obtain Nevanlinna type of results for curves and surfaces), algebraic geometry (where we combine Ideas of Nevanlinna theory with Hilbert problem 16). In all these fields we obtain novel type analogs of famous Nevanlinna deficiency relation.Thus some novel bridges are established between different parts of mathematics which clearly lead to a number of new projects. We mention here two of them. The one concern brand new ""universal version of value distribution theory"" which is valid for arbitrary meromorphic function in any domain. This will be applied to complex differential equations to study corresponding solutions in arbitrary domains: (very important in applications). Another one concerns differential geometry: we are about to construct Nevanlinna theory for large classes of surfaces including those of minimal surfaces."