A classic task in Computer vision is finding correspondences between geometric shapes, often represented via triangle graphs. Algorithms achieving state of the art performance on this task built on two key ingredients: A) The generation of informative local (geometry-capturing) features. B) A global transfer scheme between different surfaces, often utilizing canonically defined objects (such as the Laplace Beltrami operator). The first goal of the proposed PhD project is the production of advances in both domains by utilizing hitherto neglected concepts from topology, geometry and mathematical physics. In a second step, noting the possibility to describe bound quantum states such as molecules in geometric terms (e.g. via graphs or Connolly surfaces), the developed methods are meant to be applied to problems on graphs as they e.g. arise in drug design and molecular structure prediction.