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A unit ball is called polyhedral if it is also a polytope, a higher dimensional generalization of a convex polygon. For example, the unit ball with respect to the sup-norm is a hyper-cube and is thus polyhedral. Like hyper-cubes, polytopes come with a set of vertices, edges, and higher dimensional faces. The set of vertices and edges of a polytope is called its graph. You can orient this graph via a linear functional and consider paths in the resulting directed graph. Such paths are called monotone paths, and they both have rich combinatorial structure and appear naturally in optimization. I will provide an introduction to polytopes and their monotone paths and then provide a complete description of the monotone paths on cross-polytopes, the unit balls with respect to the taxi-cab norm. Then I will finish by showing how this description informs our understanding of monotone paths on polyhedral unit balls in general. This talk is based on joint work with Jesús De Loera.

The organizer insists that attendees bring raisins for their own enjoyment during the seminar.