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The Banff International Research Station will host the "Discrete Subgroups of Lie Groups" workshop in Banff from December 8, 2019 to December 13, 2019. Group theory is the mathematical study of objects that have a symmetric or repeating structure. The workshop will bring together researchers who have a variety of approaches to the investigation of "thin" groups, which are groups of a particular type that are not yet well understood, but have important connections to other fields of mathematics, including arithmetic, geometry, algebra, and dynamical systems (the mathematical study of motion). Lectures and work in small groups will be organized to forge new connections between the different points of view, and make progress on fundamental open questions. The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada-s Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta-s Advanced Education and Technology, and Mexico-s Consejo Nacional de Ciencia y Tecnología (CONACYT).