Explore the intricacies of face structures in tropical polyhedra through this 29-minute lecture by Ben Smith at the Hausdorff Center for Mathematics. Delve into the challenges of applying classical face definitions to tropical polyhedra, including issues with boundary tiling and vertex characterization. Examine the concept of monomial tropical polyhedra as a potential solution, arising from the tropicalization of blocking polyhedra. Learn about their unique minimal halfspace description and how it extends to a canonical halfspace description for general tropical polyhedra. Investigate possible face structures and combinatorics of tropical polyhedra, covering topics such as external descriptions, images of Puiseux polyhedra, geometry of tropical faces, and internal descriptions of monomial tropical polyhedra. Gain insights into this collaborative research with Georg Loho, aimed at developing a coherent notion of tropical facets and advancing the understanding of tropical polyhedra's combinatorial properties. Read more