Project Title: Delta Gordian Distance Between Links

Project Directors: Anthony Bosman, Yun Myung Oh

Project Summary: Through an 8-week summer project running throughout June and July 2022, we seek to support students from diverse backgrounds to experience how mathematics is done and see themselves as researchers. The first weeks will introduce them to knot theory, with special attention to a diagrammatic move known as the Delta move and several related knot and link invariants. Then each student will pursue a research project exploring the Delta-Gordian distance, that is, the minimal number of Delta moves between two links. Their work will result in original results which they will learn to write up for publication and present at a variety of national and regional conferences. This is a continuation of our successful 2021 NREUP at Andrews University. Directed by Anthony Bosman and co-directed by Yun Myung Oh, the program will aid students in their transition to upper-level courses and prepare students for graduate school and careers in mathematics.

Project Title: DePaul University Summer 2022

Project Directors: Karl Liechty, Emily Barnard

Project Summary: The 2022 NREUP at DePaul University will focus on the relatively new field of dynamical algebraic combinatorics. Broadly speaking, problems in dynamical algebraic combinatorics investigate enumerative questions about the orbit of a combinatorially defined map on some algebraic object. We will focus on two key maps on the symmetric group: the pop-stack sorting map and the kappa map. A typical question is: For an invertible map, is the average of ``size'' of each orbit the same? Our students will build on existing results about the behavior of these maps for the set of all permutations and for permutations which are 312-avoiding to study the maps on other pattern-avoiding permutations. Students will leave with a toolbox for exploring research questions in a variety of interconnected mathematical fields, including dynamical systems, extremal, algebraic and geometric combinatorics, and possibly topology and representation theory.

Project Title: Howard's NREUP REU+ Program

Project Directors: Dennis Davenport, Moussa Dombia

Project Summary: An important goal of this program is to encourage Howard University students from underrepresented groups to compete and succeed in the mathematical sciences. The students we plan to admit are not those who would normally be admitted to a typical REU program, in that their GPAs may not be exceptional (the average GPAs for students in the two REU programs the PI directed were all greater than 3.5) and the students may not have taken a proof-based course, which is a requirement for most REU programs. The program seeks undergraduate first and second-year students who have completed at least Calculus II with distinction and have at least a 3.0 GPA. As a follow-up, each student will be required to enroll in the 3-credit hour course UG Research Mathematics (Math 175) in the Fall of 2022 to continue their research and refine their research articles.

Project Title: Interdisciplinary Research in Graph Theory and Applications in Social Networks

Project Director: Aihua Li

Project Summary: This is a new proposal. The PI seeks an MAA NREUP grant to host an NRUEP site in summer 2022 at Montclair State University (MSU). MSU is a Hispanic Serving Institution. The PI will select four undergraduate minority students, one from MSU, two from nearby community colleges, and one from a historical black institution out of New Jersey, who have taken calculus I and/or linear algebra for this 2022 cohort. The two main projects are (1) Zero Divisor Graphs of Certain Matrices Modulo a Prime Number; (2) Study of Graph Connectivity and its Applications in Analyzing Data of a Social Network. The program emphasizes an interdisciplinary approach through both theoretical and applied research. It offers the participants opportunities to explore selected graph theory problems in a team work setting. Students will experience original mathematics research and its applications in social networks.

Project Title:Investigation of Novel Numerical Schemes for Approximating Ordinary and Partial Differential Equations

Project Directors: Treena Basu, Ron Buckmire

Project Summary: Students participating in NREUP-Occidental College will give students the opportunity to work on research problems that lie in the field of Applied Mathematics, specifically at the intersection of Differential Equations and Numerical Analysis. More specifically, in this project we will be developing and exploring non-standard finite difference (NSFD) schemes for producing numerical solutions for ODEs and PDEs. Since there are very few instances in which closed-form analytical solutions for ODEs and PDEs can be found, numerical means often have to be used, especially for differential equations describing real-world phenomena.

Project Title:Generalized Splines at SHSU

Project Director: Naomi Krawzik

Project Summary: Students participating in the National REU Program at Sam Houston State University (SHSU) will work on projects in a burgeoning topic in algebra known as generalized splines, which combines tools and knowledge from commutative ring theory and graph theory. During the summer, students will explore the ways specific graph operations affect the structure of the ring of generalized splines. In addition to their research, students will engage in other complementary activities designed to support student well-being throughout the program and beyond. In particular, students will attend two research talks, a panel discussion on graduate school in the mathematical sciences, and two presentations given by mathematicians in industry. They will also learn to use LaTeX, write a technical report on the progress of their research, and create a poster of their results. During the fall semester, students will present their work at the SHSU Undergraduate Research Symposium and at other regional conferences.

Project Title: CMAT- Computational Mathematics at Tarleton

Project Directors:Tom Faulkenberry, Scott Cook, Christopher Mitchell

Project Summary: Our NREUP project is called Computational Mathematics at Tarleton (CMAT). With this collaborative, cross-disciplinary project, we aim to continue stimulating intellectual curiosity and developing transferrable research skills in a group of 4 underrepresented minority students from the north central Texas region. The project co- directors will engage the students in an intensive summer research experience, where students will complete collaborative research projects in the field of computational mathematics. The results of this research will not only contribute to the body of scientific knowledge in these fields, but more importantly contribute to the development of these students' knowledge and research skills related to mathematics and computational science. We hope that the experience will inspire these students to persist to graduation, pursue further STEM-related educational opportunities, and ultimately seek careers in the mathematical sciences.

Project Title: Mathematics of Consumer Litter Distribution Along Metropolitan Waterways

Project Directors:Emily Hendryx, Matthew B. Parks

Project Summary: The goal of this project is to provide students with hands-on experience in applying mathematics and statistics to the real-world problem of consumer litter accumulation along local streams. In this 7-week program, students will not only develop mathematical models and perform statistical analyses based on local litter data, but they will also participate in the design and implementation of the data-collection process. Students will therefore gain authentic experience with experiment design, field work, data wrangling, basic programming skills, statistical analyses, and mathematical modeling through differential equations. Throughout the program, students will also participate in various professional development workshops, including honing written and oral communication skills for the academic/professional setting. Involvement in this program will ground students in interdisciplinary research, highlighting the role that mathematics can play in studying questions relevant to their own community and the world beyond.

Project Title: Summer Undergraduate Research Experiences at UOG (SURE@UOG)

Project Directors: Hyunju Oh, Hideo Nagahashi, Raymond Paulino, Vince Campo

Project Summary: For a 7-week period, we will engage 6 Pacific Islander undergraduate students from the University of Guam (UOG) in research projects, game theroy and coding theory. We will introduce the students to the fundamental game-theoretical concepts such as Nash equilibria and evolutionarily stable strategy and teach them how to use computational tools (Matlab), as well as analytical tools (optimization, differential equations, and linear algebra) to identify such strategies in real game theoretical models with applications in medicine - “vaccination games” where individuals have to make decisions whether to protect themselves from infectious diseases by taking costly actions such as taking a vaccine. In coding theory, they will encounter various examples of codes such as Hamming, BCH, and Reed-Solomon codes. Students will learn the fundamental concepts of coding theory such as correcting and detecting errors, information rate and distance of codes. Cryptography is the study of techniques for secure communication between a sender and receiver, in the presence of an adversary. Participants will have a choice of two topics during the program. The students of both groups will be trained in all aspects of research, starting with the ethics code, going through the workshops on using library and online resources and ending with training in delivering oral presentations as well as in using LaTeX to write mathematical papers. We expect that each student will submit at least one research paper and present their findings at least 2 seminar/conferences (including UOG).