April, 2024 (Mathematical Association of America Wisconsin Section Meeting) (Beamer Slides)
This was a talk given at the Wisconsin Section of the MAA meeting in Spring 2024 in Whitewater, Wisconsin. This talk entitled On Antimagic Labeling of Zero Divisor Graphs was about antimagic labelings of zero-divisor graphs and presented by a team of student researchers ( Stephen Cornelius, Jackson Feggestad, and Kyler Nikolai) from an independent study course with results about a forthcoming article.
April, 2023 (Mathematical Association of America Wisconsin Section Meeting) (Beamer Slides)
This was a talk given at the Wisconsin Section of the MAA meeting in Spring 2023 in Menomonie, Wisconsin. This talk entitled Cordial Labeling of Zero Divisor Graphs was about cordial labelings of zero-divisor graphs and presented by a team of student researchers ( Will Canter, Nathan Dreher, and Gabe Trebelhorn) from an independent study course with results about a forthcoming article.
April, 2023 (Mathematical Association of America Wisconsin Section Meeting)
This was a talk given at the Wisconsin Section of the MAA meeting in Spring 2023 in Menomonie, Wisconsin. This talk entitled Variations of Magic Labelings on Zero Divisor Graphs and covered magic type labelings of zero-divisor graphs and presented by a team of student researchers (Jackson Feggestad, Jacob Halvorson, Noah Royce, and Nathaen Wanta) from an independent study course with results about a forthcoming article.
February, 2022 (Seminario Interuniversitario de Investigación en Ciencias Matematicas) (Beamer Presentation)
This was a talk given virtually at the Seminario Interuniversitario de Investigación en Ciencias Matematicas in Puerto Rico. In this talk, we present some recent research on graceful labeling zero-divisor graphs of a commutative ring R with 1 not equal to 0. The zero-divisor graph Γ(R) is the undirected graph whose vertex set is comprised of the nonzero, zero-divisors of R and whose edge set is given by the relationship: distinct vertices x and y are adjacent in Γ(R) if and only if xy = 0. This graph comes up naturally when thinking about factorization properties in commutative rings with zero-divisors. Classically, the question was about the coloring number of zero-divisor graphs, but here we look at a different type of labeling called graceful labeling. We are able to find infinite classes of rings which are graceful and infinite classes of rings which are not graceful. We then turn our attention to the question of which rings admit graceful zero-divisor graphs beginning with all zero-divisor graphs on up to 14 vertices. We conclude by posing a few questions and thinking about potential for future research.
July, 2021 (Conference on Rings and Polynomials in Graz, Austria) (Beamer Presentation)
This was a talk given virtually at the Conference on Rings and Polynomials in Graz, Austria which was joint work with Jason R. Juett. The abstract was: In this talk, we give an overview of a recent paper, [1], which analyzes many ways in which unique factorization domains can be generalized to rings with zero-divisors. Numerous authors have extended this notion in various ways throughout the literature, so this article seeks to determine which notions are equivalent and determine the relationship between these various definitions. In most cases, we are able to provide a structural description of these different classes of unique factorization rings. We present these definitions of unique factorization rings, demonstrate the diagrams which explain the relationships between these possibilities, as well as (if time permits) give some samples of the structural descriptions of some of these classes of rings.
October, 2018 (AMS Central Section Meeting) (Beamer Presentation)
This was a talk given at University of Michigan AMS Fall Central Section meeting. The abstract was: In this talk, we give a brief overview of some of the results of our new paper of the same title. We study the factorization of ideals of a commutative ring using the framework of U-factorization introduced by Fletcher. This leads to several “U-factorability” properties weaker than unique U-factorization. We characterize these properties, determine the implications between them, and give several examples to illustrate the differences. If time permits, we also examine how these “U-factorability” properties behave with respect to several ring-theoretic constructions.
March, 2018 (AMS Central Section Meeting) (Beamer Presentation)
This was a talk given at The Ohio State University AMS Spring Central Section meeting. The abstract was: In this talk, we discuss edge graceful labelings of zero-divisor graphs. This continues a previous study in which graceful and harmonious labelings were investigated. These graphs arise naturally out of commutative rings with zero-divisors where the vertex set is the collection of non-zero zero-divisors and there is an edge between distinct vertices x, y ∈ Z(R) ∗ if xy = 0. These graphs are simple, undirected, connected graphs which makes them very nice candidates for labeling problems. We are able to find infinite classes of rings which admit an edge graceful labeling as well as infinite classes of rings which have no edge graceful labeling. If time permits we will also show tables of which zero-divisor graphs on a small number of vertices are edge graceful for all zero-divisor graphs up to size 14. This is a natural place to begin trying to exhaustively answer the question for all possible commutative rings with finite zero-divisor graphs.
March, 2017 (Invited Talk - AMS Southeast Section Meeting) (Beamer Presentation)*(joint work presented by Jason R. Juett)
Module Cancellation Properties. Special Session on Factorization and Multiplicative Ideal Theory - AMS Southeast Sectional Meeting, Charleston, South Carolina (March 2017). This is joint work with Jason R. Juett and Daniel D. Anderson. Our work was accepted for Jason Juett to give a twenty minute talk on our research in a special session in Factorization and Multiplicative Ideal Theory organized by J. Coykendall, E. Houston, and T. Lucas. Abstract - Over the years, several different cancellation properties of modules have been studied. For example, a module A over a commutative ring R is (resp., is restricted, is weak, is half, is half weak) (quasi-)cancellation if IA = JA ⇒ I = J (resp., IA = JA 6= 0 ⇒ I = J, IA = JA ⇒ I + (0 : A) = J + (0 : A), A = IA ⇒ I = R, A = IA ⇒ I + (0 : A) = R) for all (finitely generated) ideals. A module is (half) join principal if every homomorphic image is (half) weak cancellation. A particularly interesting question is which commutative rings have every nonzero (finitely generated) ideal (resp., module) satisfying some cancellation property. This presentation will review some basic facts and classic theorems on these topics, and then present several new results.
March, 2017 (Invited Talk - AMS Southeast Section Meeting) (Beamer Presentation)
On Labelings of Zero-Divisors Graphs. Special Session on Factorization and Multiplicative Ideal Theory - AMS Southeast Sectional Meeting, Charleston, South Carolina (March 2017). Invited to give a twenty minute talk on my research in a special session in Factorization and Multiplicative Ideal Theory organized by J. Coykendall, E. Houston, and T. Lucas. Abstract - We study graphs associated with a commutative ring with zero-divisors, R, called the zero-divisor graph, Γ(R). This is the simple, undirected graph whose vertices are the non-zero, zero-divisors and has an edge between distinct vertices x and y if xy = 0. Much of the initial research surrounding these graphs revolved around coloring the zero-divisor graph, which can be viewed as a particular type of vertex labeling. In this talk, we focus on other important labelings that have received significant attention in graph theory. We present several results about infinite classes of rings whose zerodivisor graphs either do or do not satisfy these various labeling properties. We also discuss current and future related research.
September, 2014 (Invited Talk - AMS Central Section Meeting) (Beamer Presentation)
τ-Complete Factorization in Commutative Rings with Zero-Divisors. Special Session on Commutative Ring Theory - AMS Central Sectional Meeting, Eau Claire, Wisconsin (September 2014). Invited to give a twenty minute talk on my research in a special session in Commutative Ring Theory organized by Michael Axtell and Joe Stickles. We discuss another method of extending τ- factorization to commutative rings with zero-divisors. Instead of specifying the type of element to factor into, you study the whole factorization and say it is τ-complete if it cannot be refined into a longer factorization. This yields many interesting relationships between previously studied τ-finite factorization properties in rings with zero-divisors.
March, 2014 (Invited Talk - AMS Southeastern Section Meeting) (Beamer Presentation)
On Irreducible Divisor Graphs in Commutative Rings with Zero-Divisors. Special Session on Commutative Ring Theory (in honor of the retirement of David E. Dobbs) - AMS Southeastern Sectional Meeting, Knoxville, Tennessee (March 2014). Invited to give a twenty minute talk on my research in a special session in Commutative Ring Theory organized by David Anderson and Jay Shapiro. We discuss extending irreducible divisor graphs to rings with zero-divisors. We are able to get many alternate characterizations of finite factorization properties by looking at graphs associated to commutative rings.
January, 2014 (Joint Mathematics Meetings - AMS Special Session on Trends in Graph Theory)(Beamer Presentation)
A Connection Between Graph Theory and Commutative Rings. Joint Mathematics Meeting, Baltimore, Maryland (January 2014). This is a twenty minute talk on the overlap between commutative algebra and graph theory. We begin by discussing the zero-divisor graph program initiated by I. Beck (1988) and some of the key results regarding zero-divisor graphs. Then we discuss other generalizations and analogues that have been developed, associated zero-divisor graphs, comaximal graphs, and irreducible divisor graphs. I conclude by demonstrating a few of the nice results and characterizations of finite and τ-finite factorization properties in rings.
April, 2013 (Invited Talk -AMS Central Section Meeting)(Beamer Presentation)
τ-U-Factorization in Commutative Rings with Zero-Divisors. AMS Central Section Meeting - Iowa State University - Ames, Iowa. Invited to give a twenty minute talk on my research in a special session in Commutative Ring Theory organized by Michael Axtell and Joe Stickles.
January, 2013 (Beamer Presentation)
τ-Factorization in Commutative Rings with Zero-Divisors. Joint Mathematics Meeting - San Diego, California. This talk covers many of the results from two papers covering τ-factorization and τ-U-factorization.
August, 2012 (Beamer Presentation)
Generalized Factorization in Commutative Rings with Zero-Divisors. Mathfest - Madison, Wisconsin. Gave a ten minute presentation using beamer in the general contributed paper session. The presentation is included; however, one should note there is additional material included after the "Thank You slide" which was not included in the presentation due to time constraints.
August, 2007
Constructing Families of Hadamard Difference Sets in Groups of Order 144. With Nicole Kroeger. Young Mathematians Conference - The Ohio State University. A talk given on our research from the Central Michigan NSF-REU.
January, 2008
Hadamard Difference Sets in Groups of order 144 and the Spread Construction.Joint Meetings - San Diego, CA. This was a twenty minute talk given about the research conducted over the Summer of 2007 with Nicole Kroeger, Marcus Miller, and Kathleen Shepard as part of an NSF REU at Central Michigan University.
July, 2007
Hadamard Difference Sets in Groups of order 144 and the Spread Construction.Michigan Undergraduate Mathematics Conference - Grand Valley State University, Grand Rapids, MI. This was a twenty minute talk given about the research conducted over the Summer of 2007 with Nicole Kroeger, Marcus Miller, and Kathleen Shepard. The work was the result of an NSF REU at Central Michigan University.