# Department of Mathematics and Decision Sciences Seminars

## Fall 2023

Location: Tuesdays in UC416 and Wednesdays in UC327

**Wednesday, September 6th, 3:30-4:30 pm (UC 327)**

**7-cordiality of trees,**Keith Driscoll, Clayton State University- Abstract: In 1991, M. Hovey defined the k-cordial labeling of a graph as a function from the set of vertices to Zk so that each label appears on at most one more vertex than any other, and each induced edge weight (found by summing the labels on the incident vertices, modulo k) appears on at most one more edge than any other. He conjectured that for any positive integer k, all trees are k-cordial, and showed this holds for 3 ≤ k ≤ 5. Driscoll, et. al., proved that all trees are 6 cordial. In this work we discuss the problem of proving that all trees are 7-cordial.

**Wednesday, September 27th, 3:30-4:30 pm (UC 327)**

**A discussion of functional identities,**Louisa Catalano, Clayton State University- Abstract: In this talk, we will be discussing a variety of problems in the area of functional identities. We will talk about what it means to be a functional identity, recent results, and some of the techniques used to prove these results. The talk will be accessible to any student who has seen linear algebra. Joint works with Tomás Merchán, Megan Chang-Lee, Sam Hsu, and Regan Kapalko

**Tuesday, October 3th, 3:30-4:30 pm (UC 416)**

**Knot Theory on the nxnxn Rubik’s Cube – Progress and Future Directions,**David Plaxco, Clayton State University- Abstract: In this talk, I will outline my progress in investigating the possibilities of embedding knots on nxnxn Rubik’s cubes. I will also introduce and discuss future directions for this play/work. This talk will include a few guidelines for embedding knots on nxnxn Rubik’s cubes, several photos of knots I have put on cubes, a live demonstration of how to solve a cube so that it has an embedded knot, and a setup for two Capstone projects currently underway by Seniors Adaija Warren and Alia Davis.

**Wednesday, October 11th, 3:30-4:30 pm (UC 327)**

**Smuggling boxes in shipping containers,**Everett Sullivan, Clayton State University- Abstract: In this talk, we consider packing as many boxes inside a rectangle as possible. The restriction we explore in this packing is that when a slice, either horizontal or vertical, is taken from the box, that the number of boxes in the slice is bounded by some number. We explore the maximum number of boxes that can be packed given the bounds on how many boxes may be in a slice. We will consider a range of box shapes from convex polygons to rectangles and how their geometry affects the maximum packing.

**Wednesday, October 18th, 3:30-4:30 pm (UC 327)**

**Creating an Undergraduate Research Project,**Christopher Raridan, Clayton State University-
Abstract: Undergraduate research is a high impact practice that has been shown to improve student interest in the subject matter. In this talk, we will discuss some of the steps that I have taken to create successful undergraduate research projects. We will conclude the talk by playing a game that is based on some undergraduate math research projects that were completed by students at Clayton State University.All faculty, staff, and students are encouraged to attend. No prior mathematical knowledge is required.

**TALK CANCELED**

**Wednesday, October 25th, 3:30-4:30 pm (UC 327)**

**Arithmetic geometry and stacky curves,**Andrew Kobin, Emory University- Abstract: Solutions to many problems in number theory can be described using the theory of algebraic stacks. In this talk, I will describe a few Diophantine equations, such as the ''generalized Fermat equation'' Ax^p + Bx^q = Cz^r, whose integer solutions can be found using an appropriate stacky curve: a curve with extra automorphisms at prescribed points. Time permitting, I will also describe other applications of stacky curves in number theory. Parts of the talk are joint work in progress with Juanita Duque-Rosero, Chris Keyes, Manami Roy, Soumya Sankar and Yidi Wang, and separately with David Zureick-Brown.

**Wednesday, November 8th, 3:30-4:30 pm (UC 327)**

**An Introduction to x-homotopy,**Tien Chih, Oxford College of Emory University- Abstract: Homotopy theory is broadly speaking about the continuous transformations from functions into functions or shapes into other shapes. How does this idea translate to a discrete setting like graphs? One approach would be to treat graphs as a one dimensional simplicial space, but this is not very interesting or satisfactory. Instead we ask, what would a homotopy developed natively within graphs look like? In this talk, we describe a categorically defined homotopy theory for graphs called x-homotopy, show that this allows graphs to be a 2-category with a well defined homotopy category, and discuss other results about x-homotopy and current work, including undergraduate research. This talk is accessible for undergraduates.

## Spring 2023

Location: UC 327

**Friday, February 24th, 12:00-1:00pm**

**Mathematics Colloquium: Michael Wigal, Georgia Institute of Technology**

- Abstract: We will show that if
*G*is a*2*-connected subcubic graph on*n*vertices with*n*vertices of degree_{2}*2*, then*G*has a*TSP*walk of length at most*(5n+n*, establishing a conjecture of Dvorák, Král, and Mohar. This upper bound is best possible. In particular, there are infinitely many subcubic (respectively, cubic), graphs whose minimum_{2})/4 -1*TSP*walks have length*(5n+n*(respectively,_{2})/4 -1*5n/4 -2*). We will also outline how a quadratic-time algorithm can be obtained from the proof, thus yielding a*5/4-*approximation algorithm for graphic*TSP*of simple cubic graphs, improving upon the previous best approximation guarantee of*9/7*. - Joint work with Youngho Yoo and Xingxing Yu

**Monday, February 27th, 12:00-12:45pm**

**An introduction to image deblurring and iterative regularization methods**, Lucas Onisk, Department of Mathematics, Emory University- Abstract: What’s so interesting about image deblurring? It turns out that the mathematics
behind solving the image deblurring problem can extend far beyond your phone’s camera
to applications including medical imaging (think CT

scans and MRIs), bar code scanning, non-destructive testing in science and engineering, and many more. In this talk we will begin by introducing the mathematical model behind the image deblurring problem and why the solution

may be extremely sensitive to perturbations caused by error in the available data. We will then introduce methods involving regularization and show that they can be effective in reducing this difficulty. - Student attendance is encouraged as the talk will be accessible to those with a background in linear algebra.

## Fall 2022

Location: UC 327

**Monday, August 15 ^{th}, 12:00-12:45pm**

- Organizational meeting

**Tuesday, August 23 ^{rd}, 12:00-12:45pm**

- Dr. David Plaxco, Clayton State University
- Abstract: Dr. Plaxco will discuss his experience traveling to the Bridges and MOVES conference during Summer 2022. This travel was funded by the President’s Research and Creative Endeavors Mini Grants initiative. In addition to a brief overview of the conferences, Dr. Plaxco will share the work that he presented at the conferences: Knot Theory on the nxnxn Rubik’s Cube.

**Monday, August 29 ^{th}, 12:00-12:45pm**

- Working seminar in discrete mathematics led by Elliot Krop. We will discuss Vizing's conjecture about the domination of Cartesian products of graphs and go over the proof of the result of Bartsalkin and German from 1979.

**Tuesday, September 6 ^{th}, 12:00-12:45pm**

- Dr. Dmitriy Beznosko, Clayton State University
- Abstract: What is Common Between Random Numbers and Cosmic Rays? In today’s world, random numbers are used in the multitude of the fields and areas. With that widespread use, the generation of the numbers starts to be an important question as the quality of the randomness becomes of the utmost importance. After an overview of the principles behind some generation methods, the example of the use of random numbers in high energy physics will be reviewed.

**Monday, September 12 ^{th}, 12:00-12:45pm**

- Working seminar in discrete mathematics led by Elliot Krop. We will continue our discussion of Vizing's conjecture about the domination of Cartesian products of graphs.

**Tuesday, September 20 ^{th}, 12:10-12:55pm **

- Working seminar on generating random matrices led by Dr. Brianna Vick.

**Friday, September 30 ^{th}, 12:00-12:45pm **

- Working seminar in discrete mathematics led by Elliot Krop. We continue our discussion of Vizing's conjecture, and will focus on the 2016 paper by Aziz Contractor (Clayton State student) and Elliot Krop that extended the result of Bartsalkin and German to broader classes of graphs. A copy of the paper can be found here.

**Tuesday, October 4 ^{th}, 12:00-12:45pm **

**Self-contracted****curves**, Tomás Merchán , Clayton State University- Abstract: A curve Γ : [0, 1] → R
^{n}is called self-contracted if for any 1 ≥ r ≥ s ≥ t ≥ 0, Γ satisfies that |Γ(r) − Γ(s)| ≤ |Γ(r) − Γ(t)|. In the talk we will study this class of curves and we will go over some recent results. - The talk will be accessible to undergraduate students!

**Monday, October 10 ^{th}, 12:00-12:45pm**

- Fall Break--No talk

**Tuesday, October 18 ^{th}, 12:00-12:45pm**

- No talk

**Monday, October 24th, 12:00-12:45pm**

- Seminar in discrete mathematics on Vizing's conjecture. We make our first natural generalization of the Contractor-Krop argument. This is a working seminar, in the form of a group discussion, so there will be no lead lecturer.

**Tuesday, November 1 ^{st}, 12:00-12:45pm**

**Is SVD the GOAT?**David Williams, Clayton State University- ABSTRACT:In numerical linear algebra, one of my favorite things is the singular value decomposition of a matrix (SVD). The SVD has an enormous number of applications and every matrix has one. I will briefly discuss some basic geometry related to the SVD and indicate some of the actual complexities of calculating the SVD for any given matrix. I will then discuss some of the many applications of the SVD which will include: image processing, classification problems in machine learning, and eigenfaces. Finally, I will introduce a topic I am really interested in because it lies at the intersection of data science, dynamical systems, and numerical linear algebra...the dynamic mode decomposition (DMD). The DMD and its variants have become increasingly versatile tools since the first appearance of the DMD in 2008. The DMD and its variants promise to be an exciting area of mathematical and computer science research for the foreseeable future (and the DMD starts with an SVD!)
- This talk will be accessible to undergraduate students (and there will be cookies)!

**Monday, November 7 ^{th}, 12:00-12:45pm**

- No talk

**Tuesday, November 15 ^{th}, 12:00-12:45**

- TBD

**Friday, November 18 ^{th}, 12:00-1:00pm**

**Mathematics Colloquium: Tom Kelley, Georgia Institute of Technology**

**Thresholds for Latin squares and Steiner triple systems**, Tom Kelley, Georgia Institute of Technology- Abstract: An order-n Latin square is an
*n*×*n*matrix with entries from a set of*n*symbols, such that each row and each column contains each symbol exactly once. Suppose that*L*⊆ [_{i,j}*n*] is a random subset of [*n*] where each*k*∈ [*n*] is included in*L*independently with probability_{i,j}*p*for each*i,**j*∈ [*n*]. How likely does there exist an order-*n*Latin square where the entry in the*i*th row and*j*th column lies in*L*? This question was initially raised by Johansson in 2006, and later Casselgren and Ha¨ggkvist and independently Luria and Simkin conjectured that log_{i,j}*n/n*is the threshold for this property. In joint work with Dong-yeap Kang, Daniela Ku¨hn, Abhishek Methuku, and Deryk Osthus, we proved that for some absolute constant*C*, if*p > C*log^{2}*n/n*, then asymptotically almost surely there exists such a Latin square. We also prove analogous results for Steiner triple systems and 1-factorizations of complete graphs.

**Monday, November 21st, 12:00-12:45pm**

- No talk

**Tuesday, November 29th, 12:00-12:45pm**

- No talk