Sampling and Learning from Random Polynomials: Two Stories

This talk is motivated by probabilistic models of random monomial ideals that mirror and extend those from random graphs and simplicial complexes literatures. Our results provide precise probabilistic statements about various algebraic invariants of (coordinate rings of) monomial ideals: the probability distributions, expectations and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers.

We will tackle the following related questions: What is a systematic way, in a probabilistic-model sense, to generate binomial ideals randomly? What can be (machine) learned from such data sets? How do we 'test out the waters' to see if a problem is 'learnable'? How do we generate, share, and make available large training data sets for machine learning in computational algebra?

These topics are based on joint work with various collaborators and students and form a two-step process in learning on algebraic structures.