# May 1st Algebra Workshop

One day Algebra workshop for Schools in the South East. Georgia Southern U, Statesboro, GA May 1st, 2019

Room : Math/Physics 3314 (!)

## Speakers & Schedule

9:20 - 9:30 Welcome

9:30 - 10:30 Saeed Nasseh (Georgia Southern)

10:30 - 11:00 Coffee Break

11:00 - 12:00 Hernán Giraldo (Universidad de Antioquia, Colombia)

12:00 - 01:30 Lunch Break

1:30 - 2:30 José A. Vélez-Marulanda (Valdosta)

2:30 - 3:00 Coffee Break

## Venue

Department of Mathematics

Georgia Southern University

65, Georgia Ave - Room 3008

Statesboro, GA 30460-8093

Room : MP 3314

Directions : Map

!Parking : You can park in any spot but you need a parking pass. See registration

## Registration

Participation is free but we ask you to register for logistical purposes, to provide you with a parking pass and to be able to contact you in case of problem.

## Titles & Abstracts

Shapes of the irreducible morphisms and Auslander-Reiten Triangles in the stable category of modules over repetitive algebras,

Hernán Giraldo (Universidad de Antioquia, Colombia)

For the stable category of modules over a repetitive algebra, we show that the irreducible morphisms fall into three canonical forms: first, all the component morphisms are split monomorphisms (smonic case), second, all of them are split epimorphims (sepic case), and third, there is exactly one irreducible component (sirreducible case). Finally, we describe the shape of the Auslander-Reiten triangles by using the properties of the irreducible morphisms as above. This is a joint-work with Yohny Calderon-Henao and Jose Velez-Marulanda.

Human Interaction Proofs Based on Emerging and Multistable Images; An Application of the Theory of Representation of Algebras

In this talk, semimaximal rings known as tiled orders and some Brauer configuration algebras are used to encode and decode emerging image-repositories, such images are used in different types of human interaction proofs (HIPs) known as CAPTCHAs.

Fiber products of finite Cohen-Macaulay type,

Saeed Nasseh (Georgia Southern)

In this talk, we give a characterization of (Cohen-Macaulay) fiber product rings with finite Cohen-Macaulay type, that is, with only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules. This talk is based on a joint work with Sean Sather-Wagstaff, Ryo Takahashi, and Keller VandeBogert.

Auslander-Reiten Components containing string and band complexes over certain class of Brauer graph algebras.

José A. Vélez-Marulanda (Valdosta)

Let $\Lambda$ be a finite dimensional $\mathbf{k}$-algebra. Denote by $\mathcal{K}^b(\Lambda-\text{pro})$ the homotopy category consisting in the perfect complexes over $\Lambda$. For when $\Lambda$ is a certain type of Brauer graph algebra, we describe certain family of indecomposable objects in $\mathcal{K}^b(\Lambda-\text{pro})$ that can be described combinatorially using special words in $\Lambda$ called generalized strings and bands; we call these objects string and band complexes, respectively. Let $P^\bullet$ be either a string or a band complex over $\Lambda$ and let $\mathfrak{C}$ be the component of the Auslander-Reiten quiver of $\mathcal{K}^b(\Lambda-\text{pro})$ containing $P^\bullet$. We prove that $P^\bullet$ lives on the rim of $\mathfrak{C}$ and give explicit description of the successors of $P^\bullet$ in $\mathfrak{C}$. This is a joint-work with Hern\’an Giraldo and Ricardo Rueda-Robayo.

## Organizers

Alina Iacob aiacob@georgiasouthern.edu

Jimmy Dillies jdillies@georgiasouthern.edu