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The Banff International Research Station will host the "Convex Integration in PDEs, Geometry, and Variational Calculus" workshop in Banff from August 11 to August 16, 2019. There is vast ongoing interest in the so-called technique of convex integration in several areas of mathematics, as demonstrated by diverse recent contributions. Notably, the seemingly unrelated fields of materials science, fluid dynamics and symplectic geometry enjoyed significant advances through this method, to mention a few: flexibility of the energy-minimizing solutions in the description of shape memory alloys, the resolution of Onsager-s conjecture formulated in the realm of statistical mechanics, or the classification of overtwisted contact structures in all dimensions; all based on Gromov-s h-principle and further appropriate extensions of Nash and Kuiper-s iterative convex integration scheme developed for the classical isometric embedding problem in Riemannian geometry. This 5-day workshop will bring together experts from geometry, variational calculus and mathematical fluid dynamics to share existing knowledge as well as to open up new perspectives and collaborations across different mathematical subfields. 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).