Low-rank structures appear in an astonishing variety of scientific and engineering problems. From compressing images and videos, to reconstructing incomplete data in tomography, to accelerating the solution of large-scale differential equations, low-rank approximations offer a unifying principle: complex, high-dimensional data often live close to a much simpler, low-dimensional representation.

In this talk, we will introduce the concept of low-rankness and explain why it arises so naturally in applications. We will then explore how exploiting low-rank structure leads to dramatic gains in efficiency, both in terms of storage and computation, and present concrete examples from imaging, tomography, numerical PDEs, video recognition, and beyond. The goal is to convey the ubiquity and power of low-rank methods, and to provide an accessible overview of the mathematical ideas that make them such a central tool in modern computational science.