I use techniques from algebraic and differential topology that are inspired by questions in image analysis and high-dimensional data analysis. On the image and shape analysis side, one of the main tools I have used is Blum’s medial axis, a skeleton-like topological structure that captures the shape and geometric properties of a region. On the data analysis side, the techniques I utilize fall under the heading of applied and computational topology, a relatively new branch of mathematics that focuses on discovering global and local structure within a dataset and finding meaning in the “shape” of the data. I have worked on applications of topology to problems arising in image recognition, machine learning, and geometric data modeling. In recent work, I have focused extensively on topological characterizations and summaries for metric graphs and simplicial complexes associated to them.