Generalized Hamming Weights and Symbolic Powers of Stanley-Reisner Ideal Matroids

Explore a 47-minute lecture from IPAM's Computational Interactions between Algebra, Combinatorics, and Discrete Geometry Workshop where Michael DiPasquale from New Mexico State University delves into the relationship between generalized Hamming weights and symbolic powers of Stanley-Reisner ideals in matroid theory. Learn how the minimum distance of a code serves as a matroidal invariant and corresponds to the smallest degree of a squarefree monomial in the Stanley-Reisner ideal of the dual code's matroid. Discover the evolution of minimum distance coding theory into the weight hierarchy concept through generalized Hamming weights, and understand how these weights connect to symbolic powers of Stanley-Reisner ideals in matroid theory. Examine the mathematical relationship between successive symbolic powers and their squarefree monomials, leading to insights about the Waldschmidt constant and applications to matroid configurations in projective varieties introduced by Geramita-Harbourne-Migliore-Nagel.