Infinite hidden symmetries and integrability of N=4 super-Yang-Mills theory

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We are witnessing an increasing role of algebro-geometric methods in the worldwide study of strings, gauge theories and integrable models. Me and my collaborators have participated in this endeavors through the investigation of N=2 strings and their hidden Kac-Moody type symmetries, of topological strings on supertwistor spaces and the corresponding string field theories, of holomorphic Chern-Simons and super-Yang-Mills theories, as well as of integrable models in lower dimensions. Our studies have given us far-reaching experience with twistor methods, differential geometry, extended supersymmetry and integrability. This competence will be further employed to an improved handling of the infinite hidden symmetries of N=4 supersymmetric Yang-Mills theory.The present project intends to elaborate the twistor description of N=4 super-Yang-Mills theory and its self-dual subsector from the viewpoint of Poisson geometry, Poisson-Lie groups and dressing transformations. The consideration of N=4 super-Yang-Mills theory by twistor methods combined with the methods of Poisson geometry and Poisson-Lie groups remains yet unaddressed. We intend to fill this gap and to study hidden Poisson-Lie symmetries of N=4 supersymmetric Yang-Mills theory.