Anna Seigal - University of Oxford, Mathematics A quiver is a set of vertices and directed edges. A representation of the quiver assigns vector spaces to the vertices and linear maps to the edges. We consider data points in these vector spaces, interrelated via the linear maps. This gives a generalization of a tensor of multi-indexed data. We seek principal components of the data that are compatible with the linear maps of the quiver representation. To this end, we compute the vector space of sections, or compatible assignments of vectors to vertices, of a quiver. Principal components are solutions to an optimization problem over the space of sections. Based on joint work with Vidit Nanda and Heather Harrington.