O-minimal Structures and their Applications to Dynamical Systems, Complex analysis and Potential Theory

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In this project we want to investigate o-minimal structures and their applications to dynamical systems, complex analysis and potential theory.O-minimal structures generalize geometries of algebraic type. Important concepts and functions from analysis can be realized in such structures. Axiomatically, they are defined by excellent tameness and finiteness properties, which can be applied to problems from analysis. The axiomatic formulation takes place in the language of mathematical logic. O-minimal structures are very important in mathematical fundamental research to understand the connections between geometry, logic and analysis.In the present project, o-minimal structures shall be applied to the following areas from analysis:a) dynamical systems: questions around Hilbert's 16th problem, part 2,b) complex analysis: Riemann mapping theorem,c) potential theory: Dirichlet problem.Interesting relations between these research areas arise thereby.