Mathematics Colloquia and Seminars

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What are the typical properties of a monomial ideal? Which properties are ``unusual," and how unusual are they? Which properties frequently occur together, or almost never occur together? In recent work with my advisor, Jesus de Loera, and collaborators Sonja Petrovic, Despina Stasi, and Dane Wilburne, we introduced a probabilistic model for studying random monomial ideals. This model generalizes some well-known combinatorial models such as Erdos-Renyi random graphs. We proved the probability distributions, expectations, and thresholds for events involving monomial ideals, including: given Hilbert function, Krull dimension, and first graded Betti numbers. In additional to those results, I'll introduce some ongoing research into the typical properties of the minimal free resolutions of monomial ideals, and discuss some connections to computational complexity. Reference: https://arxiv.org/abs/1701.07130

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