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Explore the fundamental role of polyhedra in mathematics through this 58-minute lecture by Karim Adiprasito. Delve into the importance of polyhedra as a basic building block for various mathematical structures, including PL topology, toric varieties, representation theory, and arithmetic. Gain insights into how these geometric objects parametrize and organize our understanding of mathematical phenomena. Survey recent developments that offer new perspectives on combinatorics and other mathematical areas. Enhance your comprehension of the interconnectedness of different mathematical fields through the lens of polyhedra.