# Search Research Symposium Abstracts

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## The Trace Property in Preenveloping Classes

In this talk we will discuss the theory of trace modules up to isomorphism and explore the relationship between preenveloping classes of modules and the property of being a trace module, guided by the question of whether a given module is trace in a given preenvelope. As a consequence we use trace modules to characterize several classes of rings with a focus [Read More...]

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

**Authors:**Haydee Lindo, Peder Thompson

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 18, 2022; 10:15 am

## Differential Powers of Ideals

This talk will discuss the results of an REU project on differential powers of ideals. Inspired by results on ordinary and symbolic powers of ideals, we investigated the containment problem for differential powers. We also considered the asymptotic behavior as the differential power grows arbitrarily large. This is joint work with Lillian McPherson, Janet [Read More...]

**Presenter:**Jennifer Kenkel, University of Michigan

**Authors:**Jennifer Kenkel, Lillian McPherson, Janet Page, Daniel Smolkin, Monroe Stephenson, and Fuxiang Yang.

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 18, 2022; 10:40 am

## Staged Tree Models with Toric Structure

A staged tree model is a discrete statistical model encoding relationships between events. These models are realised by directed trees with coloured vertices. In algebro-geometric terms, the model consists of points inside a toric variety. For certain trees, called balanced, the model is known to be exactly the intersection of the toric variety and the [Read More...]

**Presenter:**Aida Maraj, University of Michigan

**Authors:**Christiane Görgen, Aida Maraj, Lisa Nicklasson

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 18, 2022; 11:05 am

## Multigraded regularity on products of projective spaces

Eisenbud and Goto described the Castelnuovo-Mumford regularity of a module on projective space in terms of three different properties of the corresponding graded module: its betti numbers, its local cohomology, and its truncations. For the multigraded generalization of regularity defined by Maclagan and Smith, these three conditions are no longer equivalent. [Read More...]

**Presenter:**Juliette Bruce, University of California, Berkeley

**Authors:**Juliette Bruce, Lauren Cranton Heller, Mahrud Sayrafi

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 18, 2022; 11:30 am

## Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber product rings

Consider the following properties for a commutative Noetherian local ring (R,m,k): (AR) Every finitely generated R-module M such that, for every i>0, Ext_R^i(M,M⊕R)=0 is a free module. (HW) Every torsion-free finitely generated R-module M with rank such that M⊗_R M∗ is MCM is a free module. Two conjectures related to these properties are: the [Read More...]

**Presenter:**Sylvia Wiegand, University of Nebraska - Lincoln

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 19, 2022; 8:50 am

## Geometry of Extremal Surfaces

Mathematicians have long been fascinated by the lines contained on surfaces. For example, surfaces of degree one and two contain infinitely many lines, but in degree three, a smooth surface over an algebraically closed field famously contains 27 lines. In degrees four and higher, a general smooth surface has no lines, and Segre showed that over the complex [Read More...]

**Presenter:**Janet Page, University of Michigan

**Authors:**Anna Brosowsky, Janet Page, Tim Ryan, and Karen E. Smith

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 19, 2022; 9:15 am

## Toric and tropical Bertini theorems in prime characteristic*

We generalize the toric Bertini theorem of Fuchs, Mantova, and Zannier to positive characteristic. A key part of the proof is a new algebraically closed field containing the field $k(t_1,\dots,t_d)$ of rational functions over an algebraically closed field $k$ of prime characteristic. As a corollary, we extend the tropical Bertini theorem of Maclagan and Yu [Read More...]

**Presenter:**Ashley K. Wheeler, Georgia Tech

**Authors:**Francesca Gandini, Milena Hering, Diane Maclagn, Fatemeh Mohammadi, Jenna Rajchgot, Ashley K. Wheeler, Josephine Yu

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 19, 2022; 9:40 am

## Equivariant Hilbert Series of Subspace Arrangements

To an hyperplane arrangement we can associate a combinatorial object, a matroid. Similarly, to a subspace arrangement we associate a polymatroid. Each subspace in the arrangement can be viewed algebraically as a linear ideal. We can study the product of these linear ideals by using the combinatorial data of the polymatroid. In particular, by tensoring the [Read More...]

**Presenter:**Francesca Gandini, Kalamazoo College

**Authors:**Francesca Gandini

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 19, 2022; 10:05 am

## Combinatorial multigraded book-keeping and the Ishida complex

Local cohomology is one of the several homological tools to determine whether a module is Cohen-Macaulay. In the case of modules over semigroup rings, local cohomology (at the maximal ideal) can be computed using the Ishida complex. It is standard to look at each multigraded component of the Ishida complex individually, as the corresponding cohomology [Read More...]

**Presenter:**Laura Felicia Matusevich, Texas A&M University

**Authors:**Laura Felicia Matusevich and Byeongsu Yu

**Symposium Year:**2022

**Session:**Homological and Combinatorial Aspects of Commutative Algebra

**Presentation Time:**June 19, 2022; 10:30 am

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