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Explore a 53-minute lecture where Laura Escobar from the University of California, Santa Cruz delves into the mathematical relationships between combinatorics and algebraic geometry through toric geometry at IPAM's Computational Interactions workshop. Discover how Newton-Okounkov bodies enable the application of combinatorial techniques from toric geometry to broader projective varieties. Learn about wall-crossing phenomena for Newton-Okounkov bodies, involving piecewise-linear mutation maps, and their connections to compactifications of cluster varieties. Examine how the integration of lattice collections related by piecewise-linear bijections forms a semi-algebraic object with distinct convexity and polyhedra properties. Understand how these mathematical structures can encode compactifications of affine varieties and how their geometric properties, including toric degenerations, can be interpreted through combinatorial analysis.
