With applications to computer graphics and engineering, the field of geometry processing seeks to develop a computatoinal toolkit for designing, manipulating, and understanding 3D geometry.  Classical algorithms for geometry processing are built on the foundations of the finite element method (FEM) and other techniques for discretizing and solving variational problems in geometry.  In this talk, I will explore alternative discretizations for geometry processing problems that build on modern machinery of automatic differentiation, stochastic gradient descent, and nonlinear function spaces.  Largely inspired by practical robustness and efficiency demands of practitioners in geometry processing, our work also suggests new directions for applied and theoretical research in PDE, spectral geometry, and related fields.
Joint work with Albert Chern, Ana Dodik, Mina Konaković Luković, Ahmed Mahmoud, David Palmer, Dmitriy Smirnov, Vincent Sitzmann, Oded Stein, Anh Truong, Stephanie Wang, and other members of the MIT Geometric Data Processing Group.