# Apprenticeship Program in Commutative Algebra

## Description

The Apprenticeship Program in Commutative Algebra is a workshop for senior graduate students, post-docs, and other early-career researchers. From January 20-24, 2024, small groups of participants will be guided by senior mentors to tackle a research-level problem in commutative algebra or a related field. The ultimate aim of the Apprenticeship Program is to demonstrate the process of creating an original research publication, including generating ideas, proving theorems, and writing up results. In addition, attendees will be able to participate in professional development activities during the week. Groups will continue working together after the week-long program concludes in order to complete their research projects.

**Application process and deadline: **Anyone who is interested in participating in the program must complete the online application form (for both participation and funding) by** August 15, 2024. **Please note that there is limited space in the program.

**Mentors and research topics:**

**Alex Fink:**My problems will involve the ideals of varieties of matrices constructed from tautological sub- and quotient bundles on Grassmannians, and their monomial initial ideals constructed from the matroid tautological classes of Berget--Eur--Spink--Tseng. We will consider e.g. Gröbner bases and free resolutions of these ideals, and how to describe these in terms of matroid invariants.**Tai Ha:**We shall look at properties and invariants of powers of ideals. Particularly, we shall discuss questions on the regularity of powers of homogeneous ideals, the ideal containment problem, and resurgence number and asymptotic resurgence numbers.**Allen Knutson:**Many of the varieties one meets in representation theory -- Schubert varieties, quiver loci, Bott-Samelson manifolds, Nakajima quiver varieties, bow varieties, braid varieties, affine Grassmannian slices, nilpotent orbit closures, wonderful compactifications, and on and on -- come with natural stratifications. On the algebra side, there are difficult but attackable questions concerning the defining equations of the strata. On the combinatorial side, affine charts on these strata sometimes have interesting Gr\"obner degenerations to the Stanley-Reisner schemes of simplicial complexes.**Jason McCullough:**I plan to have my group focus on outstanding problems on free resolutions of graded ideals in a polynomial ring. In particular, we will seek bounds on Castelnuovo-Mumford regularity and/or projective dimension in some specific cases, including but not limited to:- Eisenbud-Goto inequality on regularity for toric ideals (interpreted broadly. The surface case is particularly interesting.)
- (Projectively) normal ideals with large regularity. (There are no known counterexamples to the Eisenbud-Goto Conjecture that are normal.)
- Projective dimension bounds akin to the Eisenbud-Goto inequality for regularity.

Familiarity with free resolutions, local cohomology, or toric ideals would be useful for this project. **Sonja Petrovic:**This project will be centered on the use of probabilistic models and random sampling in commutative algebra. The types of questions we will consider come in two flavors. For the first one, a starting point are models for random monomial ideals, and seeing how they have been used to prove existence of certain structures or properties in commutative algebra: which Betti numbers are zero or nonzero for a given family of ideals? Can the models be extended for toric ideals? The second type of a question is about sampling: if I am interested in computing an ideal that has so many variables and generators that I cannot actually compute them all, but I have a theorem about their structure, what are some good ways one might wish to sample from this ideal? Both types of questions require some use of software to at least explore the data we might generate for the algebraic problem.**Karl Schwede:**In characteristic zero and p there are well understood classes of singularities defined by a resolution of singularities or the action of Frobenius respectively. In fact, singularities arising in this way are known to be closely related by reduction to characteristic p > 0. For example, log canonical thresholds are known to be related to F-pure thresholds. In mixed characteristic (ie, over the p-adic integers) the question of measuring singularities is just beginning to be explored, but there are analogs of some of the well known players from positive or zero characteristic. However, few examples are known. I will propose some computational problems in mixed characteristic related to analogs of invariants such as the F-pure threshold and F-signature from positive characteristic.**Alexandra Seceleanu:**Monomial subrings of non-standard graded rings.

The d-th Veronese subring of a standard graded polynomial ring S is the ring generated by all the monomials of S of degree d. In this setting, Veronese subrings enjoy exceptionally beautiful properties: for example, they are Cohen-Macaulay and Koszul. What features do the semigroup rings generated by all monomials of degree d of a non-standard graded polynomial ring possess? We will try to answer this question with regard to properties of interest to the participants in the group.