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The Banff International Research Station will host the "Multivariable Spectral Theory and Representation Theory" workshop in Banff from March 31st to April 5th, 2019. The workshop will bring together specialists in functional anaysis, representation theory of groups, and algebraic geometry to study the interplay between multivariable spectral theory, group theory, ad algebraic geometry. The workshop will provide a forum for discussion of mathematical topics of substantial interest which have received increased atention in the last few years. The selection of invited speakers and participants will take into account the need to allow for fruitful interactions between internationally known experts and junior mathematicians, including women and members of underrepresented groups. 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.