# Morgantown Algebra Days

## Poster Presentations

There will be a poster session for graduate students during lunch break (11:50am-2:00pm) on Saturday, April 22. We encourage all participants at the conference to enjoy our catered lunch while discussing mathematics with poster presenters.

Posters should be set up by 11:50am on Saturday. We will have display boards and pins for your posters.

One poster per individual. Any posters left behind will be disposed after the conference.

To be considered for presenting a poster at this conference, you need to submit, a title and abstract for your poster on the registration form by  April 1, 2023.

## Poster Presenters

• Souvik Dey (University of Kansas)
• Monalisa Dutta (University of Kansas)
• Uyen Le (West Virginia University)
• David Lieberman (University of Nebraska - Lincoln)
• Ritika Nair (University of Kansas)
• Shravan Patankar (University of Illinois at Chicago)

## Poster Titles & Abstracts

### Souvik Dey (University of Kansas)

E-mail: souvik@ku.edu

Title: No title has been posted

Abstract: No abstract has been posted

### Monalisa Dutta(University of Kansas)

E-mail: m819d903@ku.edu

Title: No title has been posted

Abstract: No abstract has been posted

E-mail: alaclair@purdue.edu

Title: Invariants of Binomial Edge Ideals via Linear Programs

Abstract:In 2009 Herzog et al. and independently Ohtani introduced the notion of binomial edge ideal---a binomial ideal associated to a simple graph. Since their introduction there has been much research relating the combinatorial structure of the underlying graph to the algebraic structure of the associated ideal. I introduce a novel linear program that combinatorially relates to maximally packing vertex disjoint paths inside the graph. Algebraically, this linear program, its LP-relaxation, and dual relate to the binomial grade, F-threshold, log canonical threshold, and height of the binomial edge ideal. I discuss these connections in this poster presentation.

### Uyen (ENNI) Le (West Virginia University)

E-mail: hle1@mix.wvu.edu

Title: Remarks on a conjecture of Huneke and Wiegand

Abstract: Unlike vector spaces over a field, not all modules over a ring have a basis. Using an operation known as the tensor product, the Huneke-Wiegand Conjecture (HWC) seeks conditions on a module $M$ over a one-dimensional local ring such that $M$ has a basis. This research shows (HWC) is affirmative for 2-periodic (joint work with O. Celikbas, Matsui, and Sadeghi) and 4-periodic modules.

### David Lieberman (University of Nebraska - Lincoln)

Title: Generalizing Bernstein's inequality for D-modules

Abstract: Bernstein's inequality is a classic result in the realm of D-modules. The inequality puts a lower bound on the dimension of a D-module over polynomial rings. More generally, when the inequality holds modules that achieve this lower bound enjoy nice properties. Our overall goal is to show the inequality holds in certain non-polynomial ring settings. This has previously been shown in the case of Veronese rings. In this poster we present a new proof of the same fact that we hope may be useful in showing other singular rings satisfy Bernstein's inequality.

### Ritika Nair (University of Kansas)

E-mail: rnair@ku.edu

Title: On the Lefschetz property for quotients by monomial ideals containing squares of variables

Abstract: Let $\Delta$ be an (abstract) simplicial complex on $n$ vertices. One can define the Artinian monomial algebra $A(\Delta) = \Bbbk[x_1, \ldots, x_n]/ \langle x_1^2, \ldots, x_n^2, I_{\Delta} \rangle$, where $\Bbbk$ is a field of characteristic $0$ and $I_{\Delta}$ is the Stanley-Reisner ideal associated to $\Delta$. For a general linear form $\ell$, $A(\Delta)$ has the Weak Lefschetz Property (WLP) if the homomorphism induced by multiplication by $\ell$, $\mu_i : A(\Delta)_i \to A(\Delta)_{i+1}$ has maximal rank for all $i$. In the joint work with Hailong Dao, we aim to characterize the Weak Lefschetz Property (WLP) of $A(\Delta)$ in terms of the simplicial complex $\Delta$. In particular, we see a complete characterization of WLP in degree $1$ in terms of the edge graph of $\Delta$. We also look at pseudomanifolds of small dimension and some interesting results relating WLP with colorability of triangulation of the pseudomanifold. We also construct a family of Artinian Gorenstein algebras that fail WLP by combining our results and the standard technique of Nagata idealization.

### Shravan Patankar(University of Illinois at Chicago)

E-mail: spatan5@uic.edu

Title: Coherence of absolute integral closures

Abstract:We prove that the absolute integral closure of an equi-characteristic zero noetherian complete local domain R is not coherent, provided dim(R) ≥ 2. As a corollary, we give an elementary proof of the mixed characteristic version of the result due to Asgharzadeh and extend it to dimension 3. Furthermore, we apply the methods of Aberbach and Hochster used to prove the positive characteristic version of this result to study F-coherent rings and our work naturally suggests a mixed characteristic analogue of a result of Smith.

### NAME/LAST NAME (Institution)

E-mail:

Title: No title has been posted

Abstract:  No abstract has been posted