Random Algebraic Structures: Monomial Ideals and Numerical Semigroups

This thesis research proposal is in the area of probabilistic commutative algebra. The mathematical fields of probability and algebra are no strangers and have had many interactions over the centuries. In the present context, the phrase "probabilistic commutative algebra" will refer specifically to the application of tools and techniques coming from probabilistic combinatorics (the probabilistic method, and the notion of a threshold function, for example) to problems and questions arising in commutative algebra.

The motivation for this line of research stems from a very simple observation: many of the objects of study in commutative algebra come equipped with some intrinsic combinatorial structure. On the other hand, probabilistic reasoning has been employed with great success in a wide array combinatorial applications. Taken together, these two facts imply exciting prospects for the field of probabilistic commutative algebra.

The talk will focus on the following two problems: random numerical semigroups and a complex of irreducibles, and random monomial ideals. The first project proposes a simple probabilistic model of random numerical semi- groups and study the behavior of several fundamental invariants under the model: the embedding dimension, genus, and Frobenius number. We also introduce a simplicial complex associated to each integer n 2 which encodes probabilistic information about the numerical semigroup membership of n and provides a connection to the field of combinatorial commutative algebra via its corresponding Stanley-Reisner ideal. The second project is inspired by the study of random graphs and simplicial complexes, and motivated by the need to understand average behavior of ideals, we propose and study probabilistic models of random monomial ideals. We prove theorems about the probability distributions, expectations, and thresholds for events involving monomial ideals with given Hilbert function, Krull dimension, first graded Betti numbers, and present several experimentally- backed conjectures about regularity, projective dimension, strong genericity, and Cohen-Macaulayness of random monomial ideals.

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