# Search Research Symposium Abstracts

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## Resolutions of Powers of Square-free Monomial Ideals

Using combinatorial structures to obtain resolutions of monomial ideals traces back to Diana Taylor’s thesis, where a simplex associated to the generators of a monomial ideal was used to construct a free resolution of the ideal. This concept has been expanded, with various authors determining conditions under which simplicial or cellular complexes can be [Read More...]

**Presenter:**Susan Morey, Texas State University

**Authors:**Susan M. Cooper, Sabine El Khoury, Sara Faridi, Sarah Mayes-tang, Susan Morey, Liana M. Sega, Sandra Spiroff

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 10:15 am

## Subcomplexes of Certain Free Resolutions

What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this talk, I will show how one can invoke the Bernstein--Gelcprime fand--Gelcprime fand (BGG) correspondence to address this question for free resolutions given by two well-known [Read More...]

**Presenter:**Aleksandra Sobieska, University of Wisconsin - Madison

**Authors:**Maya Banks and Aleksandra Sobieska

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 10:40 am

## Symbolic Powers and Free Resolutions of Generalized Star Configurations of Hypersurfaces

As a generalization of the ideals of star configurations of hypersurfaces, we consider the $a$-fold product ideal $I_a(f_1^{m_1} ···f_s^{m_s})$ when $f_1,…,f_s$ is a sequence of generic forms and $1 ≤ a ≤ m_1 +· · ·+m_s.$ Firstly, we show that this ideal has complete intersection quotients when these forms are of the same degree and essentially [Read More...]

**Presenter:**Kuei-Nuan Lin, Penn State University

**Authors:**Kuei-Nuan Lin and Yi-Huang Shen

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 11:05 am

## Virtual criterion for generalized Eagon-Northcott complexes

The Eagon-Northcott complex of a map of finitely generated free modules has been an interest of study since 1962, as it generically resolves the ideal of maximal minors of the matrix that defines the map. In 1975, Buchsbaum and Eisenbud described a family of generalized Eagon-Northcott complexes associated to a map of free modules, which are also generically [Read More...]

**Presenter:**Caitlyn Booms, University of Wisconsin-Madison

**Authors:**Caitlyn Booms-Peot and John Cobb

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 11:30 am

## On Stable Trace Ideals and Arf Rings

In this talk we will explore the intersection of trace ideals and stable ideals, that is, ideals that are stable under homomorphisms to the ring and ideals that isomorphic to their endomorphism rings. We apply our results to the study of Arf rings. This is ongoing joint work with Hailong [Read More...]

**Presenter:**Haydee Lindo, Harvey Mudd College

**Authors:**Hailong Dao and Haydee Lindo

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 3:30 pm

## Symmetric shifted ideals and their Rees algbera

An ideal in a polynomial ring is called symmetric if it is invariant under the natural action of the symmetric group by permutation of the variables. Among all symmetric monomial ideals, the class of symmetric shifted ideals plays an analogous role as the class of stable ideals among all monomial ideals, for instance, as they admit a componentwise linear [Read More...]

**Presenter:**Alessandra Costantini, Oklahoma State University

**Authors:**Alessandra Costantini and Alexandra Seceleanu

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 3:55 pm

## Neural codes, oriented matroids, and their ideals

A combinatorial neural code describes a pattern of neural activity in terms of which neurons fire together. Many types of neurons respond to a set of stimuli, called the neuron's receptive field. The neural ideal is a pseudomonial ideal which encodes the relationships between receptive fields entailed by a neural code. Oriented matroids are combinatorial [Read More...]

**Presenter:**Caitlin Lienkaemper, Pennsylvania State University

**Authors:**Alexander Kunin, Zvi Rosen, and Caitlin Lienkaemper

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 4:20 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 which is analogous to the [Read More...]

**Presenter:**Augustine O'Keefe, Connecticut College

**Authors:**Jenifer Biermann, Selvi Kara, Kuei-Nuan Lin, Augustine O'Keefe

**Symposium Year:**2022

**Session:**Combinatorial and Homological Methods in Commutative Algebra

**Presentation Time:**June 17, 2022; 4:45 pm

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