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Conformal geometry is the study of properties of spaces and mappings which depend on the measurement of angles but not of lengths. An example is the stereographic projection mapping from a sphere to a plane, which distorts lengths but preserves angles. Mathematicians have made much progress in recent years in the study of conformal geometry by using and developing new tools and methods from a variety of mathematical fields, including analysis, non-linear PDE and topology. This workshop will bring together experts from a variety of mathematical backgrounds with the goal of furthering our understanding of conformal geometry and its complex number analogue, CR geometry.