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Partial differential equations arise ubiquitously as laws of physics and other mathematical models across the scientific and engineering disciplines. The solutions to these equations describe the behavior of the systems modeled by the equations, but very often the solutions are too complicated to be found exactly. As an example from one of many disciplines, equations governing the flow of liquids or gases can be simple to write down, but solutions corresponding to extremely intricate turbulent motions of the fluids are impossible to find. One approach in such cases is to use powerful computers to approximate single solutions. A second complementary approach is to seek mathematical statements that give only partial information about solutions but that apply broadly to all possible solutions. This Focused Research Group aims to improve and apply methods that take the latter approach. This will be done by using recently developed tools from the field of optimization to solve research questions in the field of partial differential equations. This requires bringing together researchers in both of these largely separate mathematical disciplines, which is made possible by the BIRS Focused Research Group program.