New Trends in Teichmüller Theory and Mapping Class Groups

The program “New Trends in Teichmüller Theory and Mapping Class Groups” brought together people working in various aspects of the field and beyond. The focus was on the recent developments that include higher Teichmüller theory, the relation with three-manifolds, mapping class groups, dynamical aspects of the Weil-Petersson geodesic flow, and the relation with physics. The goal of bringing together researchers in these various areas, including young PhDs, and promoting interaction and collaboration between them was attained. Mathematics Subject Classification (2010): Primary: 32G15, 30F60, 30F20, 30F45; Secondary: 57N16, 30C62, 20G05, 53A35, 30F45, 14H15, 20F65 IMU classification: 4 (Geometry); 5 (Topology). Introduction by the Organisers Teichmüller theory is a broad and important field of research, and it can be considered from various point of views (algebraic geometry, hyperbolic geometry, complex analysis, uniformization theory and partial differential equations). This stems from the fact that Teichmüller space can be seen as a space of equivalence classes of marked conformal structures, or of marked hyperbolic metrics, or of complex algebraic curves, or of representations of the fundamental group of a surface into the Lie group PSL(2,R). There are also other aspects. The theory has a wide 396 Oberwolfach Report 07/2014 range of applications in low-dimensional topology, algebraic topology, representations of discrete groups in Lie groups, symplectic geometry, topological quantum field theory, string theory, etc. The workshop New Trends in Teichmüller Theory and Mapping Class Groups, organized by Shigeyuki Morita (Tokyo), Athanase Papadopoulos (Strasbourg), Robert C. Penner (Aarhus and Caltech) and Anna Wienhard (Heidelberg) was attended by 55 participants from all over Europe, the Americas and several countries in Asia. The participants included world specialists of the subjects and also young researchers, comprising more than 10 PhD students and several post-docs. The discussions and the talks concerned several aspects of Teichmüller theory, that included complex geometry in one and in several variables (Riemann surfaces and uniformization and families of Riemann surfaces), the study of symmetric spaces and the analogies with Teichmüller theory, the metric theory (Teichmüller, WeilPetersson and Thurston), complex projective structures on surfaces, 3-manifolds and their invariants, mapping class groups (representation theory, factorizations, quasihomomorphisms, Johnson-Morita theory, characteristic classes), relation with theoretical physics (chord diagrams and random matrices), representations of surface groups in character varieties in arbitrary semisimple Lie groups modulo conjugacy (higher Teichmüller theory), dynamics (Teichmüller and Weil-Petersson geodesic flows) and the symplectic geometry of moduli spaces. There was also an open problem session. Several new collaborations were started during that workshop. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1049268, “US Junior Oberwolfach Fellows”. Moreover, the MFO and the workshop organizers would like to thank the Simons Foundation for supporting Sumio Yamada in the “Simons Visiting Professors” program at the MFO. New Trends in Teichmüller Theory and Mapping Class Groups 397 Workshop: New Trends in Teichmüller Theory and Mapping Class Groups