MathAcrossCampus is a quarterly colloquium series at the University of Washington to showcase applications of mathematics, with a special emphasis on the growing role of discrete methods in math applications. The goal of this seminar is to expose theoreticians to applied work, to create a community of mathematicians and users of mathematics at UW, and to serve as a guide to students and researchers looking for projects and jobs in math-related areas by offering exposure to ongoing math applications in the Seattle area.
We will show how physical metaphors can help us understand the structure of a graph. The graphs arising in different disciplines can have very different characteristics: social networks, protein-protein interactions networks, road networks, and scientific meshes are all graphs. But, they can look very different from each other. This diversity makes it difficult to understand arbitrary graphs.
We will explore an approach to understanding graphs that has been unreasonably successful: imagining that a graph represents a phyisical object. For example, we may pretend that the edges of a graph are springs, rubber bands, or resistors. Linear algebraic techniques for understanding these physical systems naturally lead to the development of spectral and algebraic graph theory. We will survey some of the fundamental ideas from these fields.
What do a biomedical engineer aiming to speed up MRI scans, and a bookseller aiming to recommend books to readers have in common? Both can benefit dramatically from low dimensional structures and sparsity in their underlying models. Compressed sensing, which exploits sparsity , has revolutionized signal processing and medical imaging. Low-rank matrix estimation is at the heart of the recommendation problem and other modern data analysis tasks. This talk draws a mathematical connection between these seemingly disjoint problems and several others, and shows how sparsity and structure can be exploited to recover signals and models from very limited information.
MathAcrossCampus is also made possible by the efforts of UW Mathematics graduate students Clayton Barnes, Gerandy Brito Montes de Oca, Toby Johnson, Matthew Junge, Hon Leung Lee, Richard Robinson, and Erik Slivken.
Additional support has been provided by: The NSF VIGRE grant at UW; the departments of Applied Mathematics and Economics; the Milliman Fund; and the NSF Research Training Group in Inverse Problems and PDEs.