Abstract: Recently developed methods and a deep articulation of ideas from the Calculus of Variations, Geometric Measure Theory, and nonlinear Partial Differential Equations (PDEs), have been instrumental in the mathematical rigorous understanding of nonlinear phenomena in a plethora of physical and technological applications, ranging from analyzing instabilities in novel advanced materials to the denoising of medical images. This leads to the two talks of this series: Variational Problems in Materials Science and Variational Problems in Imaging Science.