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Free resolutions for principal symmetric ideals

We introduce the class of principal symmetric ideals, which are ideals generated by the orbit of a single polynomial under the action of the symmetric group. Fixing the degree of the generating polynomial, this class of ideals is parametrized by points in a suitable projective space. We show that the minimal free resolution of a principal symmetric ideal is [Read More...]

Presenter: Alexandra Seceleanu, University of Nebraska-Lincoln
Authors: Megumi Harada, Alexandra Seceleanu, and Liana Sega
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: September 30, 2023; 2:00 pm

A Taylor Resolution Over Complete Intersections

The Taylor resolution is a fundamental object in the study of free resolutions over the polynomial ring, due to its explicit formula, cellular/combinatorial structure, and applicability to any and all monomial ideals. This talk generalizes the Taylor resolution to complete intersection rings via the Eisenbud–Shamash [Read More...]

Presenter: Aleksandra Sobieska, University of Wisconsin - Madison
Authors: Aleksandra Sobieska
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: September 30, 2023; 2:25 pm

Equivariant resolutions over Veronese rings

We give explicit, characteristic-free constructions of $\mathrm{GL}(V)$-equivariant minimal free resolutions of all isotypic components of the polynomial ring $S = \mathrm{Sym}(V)$ over its $d^{th}$ Veronese subalgebra $S^{(d)}$. These isotypic components come from an action of $\mathbb{Z}/d\mathbb{Z}$ on $S$ for which $S^{(d)}$ is the ring of invariants. [Read More...]

Presenter: Sasha Pevzner, University of Minnesota, Twin Cities
Authors: Ayah Almousa, Michael Perlman, Alexandra Pevzner, Victor Reiner, Keller VandeBogert
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: September 30, 2023; 2:50 pm

Monomial ideals in affine semigroup rings

Affine semigroup rings are algebras (over a field) generated by finitely many Laurent monomials. Such rings are very amenable to combinatorial treatment, especially methods from polyhedral geometry and integer programming. This makes them attractive as examples for computations in commutative algebra, since they can exhibit a variety of behaviors that can be [Read More...]

Presenter: Laura Matusevich, Texas A&M University
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: September 30, 2023; 3:15 pm

Residual Intersections of Modules

Motivated geometrically by the study of intersections of algebraic variety, the theory of linkage and residual intersections has plenty of beautiful applications to commutative algebra. For instance, linkage provides an interesting duality between ideals which preserves various homological properties, while residual intersections have proved to be an [Read More...]

Presenter: Alessandra Costantini, Oklahoma State University
Authors: Alessandra Costantini, Louiza Fouli and Jooyoun Hong
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 1, 2023; 2:00 pm

Linkage and F-Regularity of Determinantal Rings

We prove that the generic link of a generic determinantal ring defined by maximal minors is strongly F-regular. In the process, we strengthen a result of Chardin and Ulrich. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that they are, in fact, [Read More...]

Presenter: Yevgeniya Tarasova, University of Michigan
Authors: Vaibhav Pandey, Yevgeniya Tarasova
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 1, 2023; 2:25 pm

Associated primes of powers of monomial ideals

To every ideal $I$ in a ring one can associate a unique set of prime ideals, the so-called associated primes of $I$. In many settings, these primes can be interpreted as structure revealing building blocks of $I$. The associated primes of an ideal in $\mathbb{Z}$ correspond to the prime divisors of the generator of the ideal; the associated primes of the [Read More...]

Presenter: Jutta Rath, University of Klagenfurt
Authors: Clemens Heuberger, Jutta Rath, Roswitha Rissner
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 1, 2023; 2:50 pm

Toric ideals of weighted oriented graphs

Given a vertex-weighted oriented graph, we can associate to it a set of monomials. We consider the toric ideal whose defining map is given by these monomials. We find a generating set for the toric ideal for certain classes of graphs which depends on the combinatorial structure and weights of the graph. We provide a result analogous to the unweighted, [Read More...]

Presenter: Jennifer Biermann, Hobart and William Smith Colleges
Authors: Jennifer Biermann, Selvi Kara, Kuei-Nuan Lin, Augustine O'Keefe
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 1, 2023; 3:15 pm

Canonical forms of neural ideals

The neural ideal was introduced by Curto, Itskov, et al in 2013 to study the firing pattern of a set of neurons (called a neural code), turning problems in neuroscience and coding theory into algebraic questions. They also introduced the canonical form of a neural ideal, a set of pseudomonomial generators uniquely tied to the original neural code. In this [Read More...]

Presenter: Rebecca R.G., George Mason University
Authors: Hugh Geller, Rebecca R.G.
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 2, 2023; 8:30 am

Counting lattice points that appear as algebraic invariants of Cameron-Walker graphs

In 2021, Hibi et. al. studied lattice points in $\mathbb{N}^2$ that appear as $(\text{depth}\, R/I,\dim R/I)$ when $I$ is the edge ideal of a graph on $n$ vertices, and showed these points lie between two convex polytopes. When restricting to the class of Cameron--Walker graphs, they showed that these pairs do not form a convex lattice polytope. In this [Read More...]

Presenter: Iresha Madduwe Hewalage, Dalhousie University
Authors: Madhushika Madduwe Hewalage, Sara Faridi
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 2, 2023; 8:55 am

Galois groups of random additive polynomials

The Galois group of an additive polynomial over a finite field is contained in a finite general linear group. We will discuss three different probability distributions on these polynomials, and estimate the probability that a random additive polynomial has a "large" Galois group. Our computations use a trick that gives us characteristic polynomials [Read More...]

Presenter: Eilidh McKemmie, Rutgers University
Authors: Lior Bary-Soroker, Alexei Entin and Eilidh McKemmie
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 2, 2023; 9:20 am

Galois Extensions in Division Algebras Over Semi-Global Fields

In the last 2 decades a method of constructing algebraic objects over semi-global fields (one-variable function fields over complete discretely valued fields) by patching together compatible objects constructed on a network of field extensions has been introduced and developed by Harbater, Hartmann, and Krashen. Field patching has proven to be a powerful [Read More...]

Presenter: Yael Davidov, University of Delaware
Authors: Yael Davidov
Symposium Year: 2023
Session: Combinatorial and Homological Methods in Commutative Algebra [Organized by Francesca Gandini and Selvi Kara]
Presentation Time: October 2, 2023; 9:45 am

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