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This lecture is the first part of the Terng Lecture Course on "Log-concavity and Matroids" presented by Josephine Yu from Georgia Institute of Technology at the 2025 Program for Women+ and Mathematics hosted by the Institute for Advanced Study. Explore the fascinating world of matroids, combinatorial structures that model independence in contexts like graph edges and vector spaces. Discover the surprising connections between matroids and multivariate polynomials that exhibit log-concavity in the positive orthant. Learn about the significance of log-concavity across various mathematical fields including combinatorics, algebraic geometry, convex analysis, and optimization. Delve into the real and combinatorial geometry that underlies log-concavity and its applications to matroids and random walk mixing times in this 71-minute comprehensive introduction.
