Commutative Algebra and Algebraic Geometry: The Fellowship of the Ring: Random monomial ideals and their homological properties

Seminar | April 10 | 3:45-5 p.m. | 939 Evans Hall

 Lily Silverstein, UC Davis

 Department of Mathematics

Randomness is an important tool in algebra, especially from an algorithmic perspective. I will discuss our recent work looking at the random behavior of monomial ideals. We describe several random models, inspired by earlier models for random graphs and random simplicial complexes, and give results on properties such as Hilbert function and Krull dimension. We also prove "threshold behavior" in the model parameters. In this talk I'll focus on properties related to the minimal free resolutions of random monomial ideals, in particular projective dimension, Cohen-Macaulayness, and genericity/the Scarf complex. I'll also discuss some open problems and ongoing work.