Abstract: Many problems in geometry processing, graph theory, and machine learning involve optimizations whose variables are defined over a geometric domain. The geometry of the domain gives rise to geometric structure in the optimization problem itself. In this talk, I will show how leveraging geometric structure in the optimization problem gives rise to efficient and stable algorithms applicable to a variety of application domains. In particular, I will describe new methods for problems arising in shape analysis/correspondence, flows on graphs, and surface parameterization.