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 conjectured or unusual objects. (This last technique is sometimes called “the probabilistic method.”)
This talk will apply these techniques to commutative algebra, using monomial ideals as a bridge between combinatorics and algebra. I’ll introduce a family of random models for monomial ideals, and describe results of each type mentioned above, for instance: (1) typical projective dimension, (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 subsets of: Jesús A. De Loera, Serkan Hoşten, Robert Krone, Sonja Petrović, Despina Stasi, Dane Wilburne, and Jay Yang.