Random Monomial Ideals

Probability is a now-classic tool in combinatorics, especially graph theory. Some applications of probabilistic techniques are: (1) describing the typical/expected properties of a class of objects, (2) uncovering phase transitions and sudden thresholds in the dependence of one property on another, and (3) producing examples of extremal or conjectured objects. (This last technique is sometimes called The Probabilistic Method.)

This talk explores these techniques in a commutative algebra setting, using monomial ideals as a bridge between combinatorics and algebra. I'll introduce a family of random models for monomial ideals, and describe the algebraic properties of (quotient rings defined by) these random ideals. We have results of each type described above, for instance: (1) the typical projective dimension of K[x_1,...,x_n] mod a random monomial ideal, (2) thresholds in Krull dimension as a function of number of monomial generators, and (3) how to generate unlimited examples of monomial ideals which aren't generic (in the Bayer-Peeva-Sturmfels sense), but which nevertheless have minimal free resolutions that can be read from their Scarf complexes.

Joint work with Jesus A. De Loera, Sonja Petrovic, Despina Stasi, and Dane Wilburne (pdf: Random Monomial Ideals), De Loera, Serkan Hoşten, and Robert Krone (pdf: Average Behavior of Minimal Free Resolutions of Monomial Ideals), and ongoing work with Hoşten, Wilburne, and Jay Yang.