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Explore the foundations and cutting-edge research in discrete group actions through this advanced mathematical school and workshop organized by ICTP Mathematics. Delve into the geometric study of discrete actions of groups and semi-groups, tracing from Poincaré's pioneering work on differential equations to modern applications in complex geometry and holomorphic dynamics. Master fundamental concepts including real and complex hyperbolic geometry, Kleinian groups, discrete group actions on complex manifolds, complex affine geometry, Lorentzian geometry, holomorphic dynamics, endomorphisms of projective spaces, complex representations of fundamental groups, moduli spaces of dynamical systems and representations, topology and dynamics of character varieties, and Teichmüller theory. Engage with comprehensive lecture series covering complex dynamics and character varieties, Lorentzian geometry, holomorphic dynamics, hyperbolic surfaces and Teichmüller theory, complex hyperbolic geometry, and limit sets of discrete groups. Participate in an advanced workshop featuring presentations from leading researchers on topics such as identities on hyperbolic surfaces, limits of limit sets, Kulkarni limit sets of complex hyperbolic groups, holomorphic geometric structures on parallelizable manifolds, Poincaré extensions and cobordisms of rational functions, character varieties of fundamental groups, the Einstein Universe, volume of closed anti-de Sitter 3-manifolds, conjugation classes in SL(4,C) and SU(3,1), marked length spectral rigidity for flat metrics, non-arithmetic complex hyperbolic lattices, representations of fundamental groups, geometric structures on complex curves, commensurability of hyperbolic Coxeter groups, quaternionic Kleinian modular groups, pressure on metrics on higher Teichmüller spaces, and higher Teichmüller theory.
