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The Banff International Research Station will host the "Mutations: Mirror Symmetry, Deformations, and Combinatorics (HALF)" workshop in Banff from August 11, 2019 to August 16, 2019. Mutations, introduced by Fomin and Zelevinsky in 2001, play an important role in a broad range of mathematical areas: from theoretical physics to algebraic geometry; from deformation theory to convex geometry. The basic idea of mutation is to provide an iterative way of modifying one ``good-- mathematical object to obtain another ``good-- mathematical object. Starting from an initial seed, successive mutations allow us to obtain more and more information about the objects being studied. Although the original approach by Fomin and Zelevinsky was purely algebraic, mutations are rapidly finding their way into the broader mathematical landscape. Especially interesting is recent progress towards classifying fundamental geometric spaces called Fano manifolds. Central to this new approach, which draws on ideas from Mirror Symmetry, is a detailed understanding of mutation. Building bridges between the different viewpoints and techniques is essential if this approach is to succeed. The goal of this workshop is to draw together world-leading experts from many areas of pure mathematics who might not otherwise interact. The common theme is the presence of mutations in our work, with the focus on overlaps with Mirror Symmetry, deformation theory, and combinatorics. We will exchange information on recent developments, and work together on common problems with the aim of furthering our understanding of mutations. We believe that the connections that have been discovered so far have barely scratched the surface, and we expect novel relationships between these different areas to emerge as our knowledge grows. 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).