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National Research Experience for Undergraduates Program (NREUP)

The Mathematical Association of America (MAA) supports the participation of undergraduates in the mathematical sciences in focused and challenging research experiences, as such experiences increase their interest in advanced degrees and careers in mathematics. The MAA’s National Research Experiences for Undergraduates Program (NREUP) allows mathematical sciences faculty to apply for a grant to host a summer research program at their campus. NREUP is structured both to increase undergraduate completion rates and to encourage more students to pursue graduate study by exposing them to research experiences after they complete their sophomore year. NREUP is designed to reach students at a critical point in their career path - midway through their undergraduate programs. As coursework becomes more abstract, many promising students lose interest in the mathematical sciences. Yet with the strong connection to a faculty mentor established through the NREUP, students receive the tools they need to persevere and succeed.

Please note that the application portal has closed for the 2024 cycle.

Since 2003, through the National REU Program (NREUP), the MAA has been helping faculty recruit and mentor undergraduate research students. The program began when the MAA conducted a pilot version of this project in summer 2003 using funds from the National Security Agency (NSA) that served eight students at three sites. Since then, NREUP has grown from a pilot program to a fully developed program, having received funding from the Moody Foundation, NSA, and NSF.

Objectives

The NREUP aims to:

Create opportunities for deep and meaningful undergraduate research.

Build faculty capacity and develop a workforce that can support and mentor all students.

Create a sense of belonging and community.

Nature of the Grant

The MAA invites mathematical sciences faculty to apply for grants to host an MAA Student Research Program on their campuses for at least seven weeks during summer. The faculty member should construct a program that will provide an appropriate and supportive research environment for undergraduate students, who are usually in their first research experience. In their proposal, the faculty member should discuss how they will build student ability and interest in persisting and thriving in the mathematical sciences, by making sure they can include details of both the mathematical research content and problems and the mentoring environment. A full list of proposal requirements can be found here.

These grants will support stipends for one faculty researcher and a minimum of four local undergraduates, as well as costs for student room and board.

Support for this Program

Support for this Mathematical Association of America (MAA) program is provided by the National Science Foundation, Award #DMS-1950644.

Contact

For further information about the program, please contact MAA Programs.

NREUP 2021 Student Showcase Videos

In 2021, students who participated in an NREUP were invited to present their project in a virtual poster session. The full program of the event, including project abstracts, can be found here.

In addition to their live presentations, eight student groups chose to produce an additional two-minute video presentation of their project to be shared publicly. We are thrilled to be able to share these videos with you.

Andrews University

Berrien Springs, Michigan

The Delta-unlinking Number of Algebraically Split Links

Student Presenters: Moises Reyes

Faculty Advisor: Anthony Bosman

Embry Riddle Aeronautical University

Daytona Beach, Florida

Statistical Analysis of Hispanic Voters' Behavior Florida

Student Presenters: Kamila Soto-Ortiz

Faculty Advisor: Mihhail Berezovski

Siena University

Loudonville, New York

Guns, Zombies, and Steelhead Axes: Cost-effective Recommendations for Surviving Human Societies

Student Presenters: Ahmani Roman

Faculty Advisor: Scott Greenhalgh

Texas A&M University-Commerce

Commerce, Texas

Properties and Parameters of Codes from Unit Graphs of Zn

Student Presenters: Victor Ezem and Ashlee Story

Faculty Advisor: Padmapani Seneviratne

University of Guam

Mangilao, Guam

A Model of the Dynamics of CRB-G with a Game Theoretical Analysis of the Effectiveness of Control Measures

Student Presenters: Jovic Aaron S. Caasi, Alex Leon Guerrero, and Kangsan Yoon

Faculty Advisor: Leslie Aquino, Aubrey Moore, and Hyunju Oh

University of Guam

Mangilao, Guam

Population Dynamics of the Mariana Eight-Spot Butterfly and Parasitoid Wasps: A Compartment Model Approach

Student Presenters: Cabrini Aguon and Andrew Lu

Faculty Advisor: Leslie Aquino, Aubrey Moore, and Hyunju Oh

University of Guam

Mangilao, Guam

Modeling and Analysis of Oryctes rhinoceros Behavior

Student Presenters: Michael Cajigal, Gabriel Florencio, and Ashley Yang

Faculty Advisor: Hyunju Oh, Leslie Aquino, and Aubrey Moore

University of Guam

Mangilao, Guam

A Game Theoretical Approach to Modeling Population Dynamics Between the H. o. Marianensis, H. Anomala and H. Bolina

Student Presenters: Andrea Gutierrez, Yuan-Jen Kuo, Sean Hipolito, and Shaun Wu

Faculty Advisor: Hyunju Oh and Leslie Aquino

PREVIOUS PROJECTS

At MAA MathFest 2023, we were delighted to celebrate 20 years of NREUP. Since 2003, NREUP has had sites in 31 states and US territories, and over 700 students have participated. Below you can read about the project sites since 2014.

Project Title: NREUP--Bucknell

Project Director: Nathan Ryan

Project Summary: Participants in the NREUP--Bucknell will formulate and verify conjectures in number theory. Specifically, they will be using random matrix theory to study important features of the Riemann zeta function and other L-functions. We will accomplish this by using the computer algebra systems SageMath and PARI/GP to generate a lot of data to support conjectures that come from applying random matrix theory to the Riemann zeta function and L-functions of modular forms. Students will learn various computational techniques (e.g., computational linear algebra, high throughput computing, etc.), contribute to a mathematical database sponsored by the NSF, write papers to be submitted to mathematical or computational journals and share their results at conferences. A special feature of this NREUP is that it will be run in both English and Spanish.

Project Director: Steven Kim

Project Summary: Students are curious how mathematics and statistics are applied to real-world problems. The summer research program at CSU Monterey Bay is designed for students who have completed multivariate calculus, discrete mathematics, linear algebra, and preferably introductory statistics and some programming course. (If there is any gap in the preparations, students will be trained via a 3-week mini-course.) The common theme of the summer research program is sequential decisions using data and statistical models. Students will study and develop probability models and statistical methods for making a series of decisions based on the most updated information available to decision makers. Applied areas include, but not limited to, toxicology, kinesiology, exercise science, bioinformatics, and sports analytics.

Project Title: DePaul NREUP

Project Directors: Emily Barnard

Project Summary: Building on the success of our 2022 summer NREUP, this proposal is to renew our National Research Experience for Undergraduates Program (NREUP) at DePaul University for Summer 2023. This year we have intentionally chosen a project on graph b-colorings with questions that are low-floor and high-ceiling to accommodate students at varying places in their educational careers. Several papers have been written on this topic in recent years. Many of these papers have focused on regular graphs with large diameter, leaving many open problems related to non-regular graphs or regular graphs with small diameter. Our REU aims to address some of these questions. Since many of our students will be local to the Chicagoland area, we will also use graph theory to discuss equity issues in Chicago. Our main goal is to create opportunities for our students to conduct original research and to open the door to graduate programs and careers in mathematics.

Project Title: The Summer Undergraduate Research Experiences at LSUA

Project Director: Prakash Ghimire

Project Summary: The Summer Undergraduate Research Experiences at LSUA will allow four undergraduate students to work on research problems in the field of non-associative algebra for eight weeks in summer 23. More specifically, students will characterize the linear commuting maps of the algebra of strictly block upper triangular matrices and the linear triple centralizers of the algebra of dominant block upper triangular matrices. Students will also learn mathematics writing software LATEX and research paper writing skill. We expect students to submit at least one research paper in professional journals and present their findings at the LSUA annual scholar day and MAA Louisiana/Mississippi section meeting.

Project Title: NREUP at SUNY New Paltz: The mathematics of incorporating human behavior in epidemic modeling

Project Directors: Anca Radulescu, Moshe Cohen

Project Summary: SUNY New Paltz will conduct a 7-week summer program involving participation of four mathematics undergraduates from underrepresented groups in distinct, focused and challenging research experiences. All four projects will stem from the crucial importance to scientifically address the relationship between epidemic patterns and the population response. Research activities will focus in particular on capturing and analyzing the coupling between the pandemic dynamics and the population response, and will encompass both theoretical and data-driven modeling of this coupled system. Results of this research will inform the scientific community in its work to understand and fight current and future pandemics. Students participating in this program will gain valuable research experience with 1) traditional methods in dynamical systems; 2) statistics and big data analytics, and 3) harnessing the major impact of statistics and big data on applications. Additionally, students will receive professional development experience by preparing their work for publication and presenting their results at mathematics conferences.

Project Title: Summer Research Program on Lattice Reduction Theory

Project Director: Jingbo Liu

Project Summary: This summer research program provides a mentoring structure for historically underrepresented undergraduate mathematics students from South Texas and promotes active engagement in mathematical research on the reduction theory of lattices (mainly Lagrange reduction, Minkowski reduction, HKZ reduction, and LLL reduction) that deepens and extends certain topics covered in a standard undergraduate linear algebra course. The reduction theory of lattices is not only an essential component of contemporary algebraic number theory, but also has significant applications to lattice-based cryptography and wireless communications, among other high technology fields. The four students will study the relation between the smallest value of the parameter in the LLL algorithm which generates the shortest LLL-reduced basis vector and the determinant of the lattice, and the relation between the smallest value of that parameter in the LLL algorithm and the rank of the lattice, respectively. To accomplish this research, these students will be immersed in intensive short courses on relevant topics in modern algebra, number theory, lattice theory, and statistics, plus training in LaTeX and other software; they will also be engaged in daily experience of independent research by reading papers, giving presentations, and having group discussions. The four student researchers are expected to continue their research on the lattice reduction theory in the following school years with their mentor, to give research presentations at important regional and national conferences, and to publish the work in high/good-quality mathematical research journals.

Project Title: Mixed model implicit and IMEX Runge–Kutta methods

Project Directors: Zheng Chen, Yanlai Chen, Scott Field, Alfa Heryudono, Sigal Gottlieb

Project Summary: The focus of the proposed program is to bring together a small and focused group of undergradu- ate mathematics majors from underrepresented groups to engage in dynamic, exciting, supportive research collaborations. We will select two students from UMass Dartmouth and two students from Spelman College who will work closely with other UMassD faculty members. The research topics will build on the mixed precision Runge–Kutta methods of Z. Grant, and applied to mixed model simulations. Investigations of interest will include analytical and numerical studies of stability using different spatial discretizations. Through this program, we aim to build enthusiasm for advanced degrees and careers in mathematics, and the support of a strong peer network closely mentored and guided by faculty mentors. Through the supportive community we will build, our students will feel enabled and empowered to continue their mathematics education and explore career options in this field.

*Funding provided by the Tondeur Fund.

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).

Due to the COVID-19 pandemic, 2020 awardees were given the option to defer their funding until 2021. Those awardees are indicated with a *.

Project Title: The Summer Program In Research and Learning (SPIRAL at American)

Project Directors: Monica Jackson

Project Summary: Building upon the success of the past 17 years, the Summer Program in Research and Learning (SPIRAL) at American University aims to provide a mentoring structure for underrepresented minorities and women that promotes active engagement in mathematics through a Research Experience for Undergraduates (REU) program. The main goal is to support and encourage intercultural understanding and educational excellence. Students are being prepared to excel in the difficult field of mathematics and statistics with the goal of being capable of taking on challenging problems in the future. In addition, we are focusing on diversifying the mathematical sciences to allow for different views and experiences to help solve world problems. The 8-week onsite program will run full-time from June 1, 2020 to August 1, 2020. The research goal will be accomplished first by providing students with an intensive 3-week lecture in the field of statistics and mathematics. The students will then spend the remaining weeks actively engaged in research. Students will also participate in field fields trips to local research agencies to discover applications of mathematical and statistical research. Finally, the students will submit their work for publication in a peer-reviewed journal. By participating in SPIRAL, students in the program will interact with other students with different racial/ethnic backgrounds to understand how excellence and diversity in STEM matters.

Project Title: Effects of Strong Fusion on Links

Project Director: Anthony Bosman,Yun Myung

Project Summary: Band fusion modifies a link by fusing together two link components with a band. T. Kanenobu, D. Buck, and others have studied the effects of fusion on links and tabulated the fusion pathways between links. We extend their work with an original analysis of the effects of strong fusion. Strong fusion a modified form of fusion that preserves the number of components of a link by introducing an unknotted component around the fusion band. In particular, we link invariants--including work from U. Kaiser on dichromatic invariants--to classify which links are the result of strong fusion, tabulating all such links with up to 10 crossings. We study when various necessary conditions are sufficient to determine if a link is the result of strong fusion as well as if strong fusions occur within various families of links. Fusion and strong fusion, and more generally band surgery, are important to the study of knot and link concordance.

Project Title: NREUP-Coastal

Project Director: Aaron Yeager

Project Summary: Students participating in NREUP-Coastal will study a class of random orthogonal polynomials. More specifically, they will compute the expected number of zeros of the class of random orthogonal polynomials in the unit disk, as well as quantify the accumulation of the zeros near the unit circle. To build knowledge for the topic, the students will be immersed in intensive short courses covering relevant topics in Probability, Real Analysis, and Complex Analysis. Students will also participate in daily group discussions in order to improve comfort and confidence in communicating mathematics. Our project will produce students that are prepared to learn advanced mathematics, present mathematics more confidently, and engage with the broader mathematical community. Furthermore, we expect the students to be prepared to present their work at local, regional, and national conferences (e.g. MathFest, the Young Mathematicians Conference, etc.).

Project Title: NREUP - DSU EAGER -- Enhancing Advancement to Graduate Education through Research

Project Director: Vinodh Chellamuthu

Project Summary: In this proposed NREUP – DSU EAGER program, students will work on research problems in the area of mathematical epidemiology. Students will learn how to develop mathematical models by identifying the problem, determining the necessary assumptions, finding the interrelationships among the variables, constructing a model, developing a numerical scheme, implementing a numerical scheme in MATLAB, and interpreting the results given by their models. They will be taught LaTeX so that they can prepare papers for publication. While working on their projects, students will investigate ways to modify and incorporate environmental factors into the model. This will make the models more robust by allowing students to analyze disease transmission in more realistic settings and establish stability properties of equilibria. Students will learn advanced mathematical concepts during the research process such as modeling, ODEs, matrix analysis, and numerical analysis. They will be expected to both present their work at professional meetings and publish their findings. Students will experience an increase in awareness of the broad array of mathematical research disciplines and will leave the NREUP – DSU EAGER program being encouraged, supported and geared up to pursue graduate studies and careers in mathematics.

Project Summary: NREUP at Fairfield University is a multi-faceted seven-week program focusing on graph theory and its current applications in mathematics and other scientific disciplines. The program will prepare four undergraduate students for a career in mathematics research and offer unique opportunities for them to fully immerse in the mathematical professional and academic communities. The program will first introduce fundamental concepts in graph theory and the programming tool SAGE, following which students will be divided into smaller groups to investigate two research problems: one group will work on a random forest building process, in particular, computing the probability of obtaining given number of components in this process for various families of graphs, and the second group will work on enumerating Hamiltonian paths and cycles in various lattice graphs. The program will include professional development mentorship, including seminars talks from invited speakers, weekly discussions on career in mathematics, guidance on graduate school and financial aid applications, and conferences and travel grants. Students will write a manuscript and submit it to a journal for publication and will present their research results at national and local conferences, including a colloquium at Fairfield University. Program effectivness will be measured by tracking the trajectories of student participants as they grow in confidence and ability and monitoring REU participation, conference presentations, and graduate school applications beyond the duration of the program.

Project Title:Modeling the Spread of Infectious Diseases with Dynamical Systems

Project Directors: Matthew Johnston & Bruce E. Pell

Project Summary: The NREUP at Lawrence Technological University will introduce underrepresented minority students from the diverse Metro Detroit region to the multifaceted techniques of mathematically modeling the spread of infectious diseases. During the first two weeks of the project, students will learn the mathematical tools researchers use to estimate key epidemiological parameters, such as R_0 (R-naught), and to project the spread of diseases under a variety of public policy interventions. From the third week on, the students will independently build and analyze mathematical models for the spread of disease which account for a variety of different factors, such as demographic variations, face mask utilization, changes in social behavior, and different vaccination distribution priorities. The primary focus of the project will be model building and rigorously showing how these models behave using dynamical systems theory. A secondary focus will be on simulating the systems numerically, performing parameter estimations, and validating with real-world data.

Project Title: The evolution of zombies and cost-effective ways to prevent the end of the world.

Project Directors:Scott Greenhalgh & Kursad Tosun

Project Summary: Our summer research program focuses on applying tools from mathematical biology to predict pertinent information on the evolution and control of the undead. In particular, the program will provide insight as to whether zombies can naturally arise, and identify cost-effective strategies to control the (potentially inevitable) zombie apocalypse. Given these goals, this research program will feature two research subgroups. The first subgroup will investigate the potential for zombies to evolve from the dead through the application of an evolutionary invasion analysis on a population growth model. The second subgroup will apply cost-effectiveness analyze to a zombie outbreak model to identify economical ways to control (and potentially prevent) a zombie uprising. Students in both subgroup will be trained in all aspects of the research, starting with common techniques from applied mathematics used in the development and analysis of compartmental models, including differential equations theory, stability analysis, and probability theory. In addition, students will also enhance their computational skill sets from practical experience in coding mathematical models, and gain valuable experience both in scientific writing and in oral presentations.

Project Title: Introduction to Mathematical Modeling of Infectious Diseases

Project Directors:Enahoro Iboi & Naiomi T. Cameron

Project Summary: As a continuation of the development of the Math RaMP summer research program, we propose a seven-week summer research experience for four students in summer 2021 on the topic of modeling population-level impacts of the novel Coronavirus outbreak in the U.S. Dr. Enahoro Iboi will serve as faculty research mentor. The focus of this research experience is to encourage African American women who are mathematics majors at Spelman College to pursue graduate studies in Applied Mathematics. This NREUP will benefit students by exposing them to an area of applied mathematics (mathematical biology) not commonly presented on an undergraduate level by exposing them to modeling techniques, analysis, and simulations used for gaining insight into the transmission and control of some emerging and re-emerging diseases that are of public health concern. By the end of the program, students will be expected to be skilled in developing infectious disease models, perform basic analysis, collect, analyze, and visualize data, estimate unknown model parameters, and perform numerical simulations. The research projects will be accessible to students who have completed at least the calculus sequence and linear algebra. Thus, this program is appropriate for students who have completed their sophomore year or for students who are advanced first-year students. Students who participate in this NREUP will be expected to present at the annual Spelman College Research Day, MAA MathFest, NAM MATHFest, JMM 2022, or at a regional MAA conference and submit the results to appropriate professional journals for publication.

Project Title: CMAT: Computational Mathematics at Tarleton

Project Directors:Thomas Faulkenberry, Scott A. Cook & Christopher D. Mitchell

Project Summary: The proposed NREUP project is called Computational Mathematics at Tarleton (CMAT). With this collaborative, cross-disciplinary project, we aim to stimulate intellectual curiosity and develop transferrable mathematics 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 online 8-week research experience, where students will complete collaborative research in computational mathematics, with specific projects in mathematical modeling of cognition, billiard dynamics, and disease modeling. The results of this research will not only contribute to the body of scientific knowledge in these fields, but also contribute to the development of these students' knowledge and research skills related to mathematics and computational science, inspiring these students to persist to graduation, pursue further STEM-related educational opportunities, and ultimately seek careers in the mathematical sciences.

Project Title: Game Theory and Applications

Project Directors: Jan Rychtar, & Hyunju Oh

Project Summary: Our students will be introduced to the fundamental game-theoretical concepts (Nash equilibrium and evolutionarily stable strategy) and taught how to use computational and analytical tools to identify such strategies in models with applications in biology and/or medicine (cat vaccination to prevent Toxoplasmosis infection or bed-net use to prevent malaria). The students will be trained in all aspects of research, starting with the ethics code, going through 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.

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

Project Directors: Hyunju Oh & Leslie Aquino

Project Summary: For a 7 week period, from June 1, 2020 to July 17, 2020, we will engage 6 Pacific Islander undergraduate students from the University of Guam (UOG) in research projects. The students will work under the supervision of project directors Hyunju Oh and Leslie Aquino. Aubrey Moore and G. Curt Fiedler will provide ecology and biology background to our research groups. We will first introduce students to fundamental game-theoretical concepts, such as Nash equilibria and evolutionarily stable strategy. Then, we will teach them how to use computational tools (NetLogo and Matlab), as well as analytical tools (optimization, differential equations, and linear algebra) to identify such strategies in real game theoretical models, with applications for biology and ecology. The students 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 colloquia/conferences (including UOG).

Project Title: Deep Learning for solving Differential Equations

Project Directors: Huiqing Zhu

Project Summary: In the past decade, deep learning has become a core component of artificial intelligence and a computational technology that can be trained to automate human skills. It has a potential to transform many traditional research approaches used in science and engineering. Recently, solving differential equations via deep learning has emerged as a potentially new sub-field under the name of Scientific Machine Learning. The 2020 NREUP summer undergraduate program will provide participants (1) an introduction to machine learning and deep learning; (2) an introduction to Python; (3) hands-on practice using deep learning as a numerical tool to solve differential equations; (4) an application to predator-prey models.

Project Title:Summer Undergraduate Research Experience in Statistics at Wofford

Project Directors: Deidra Coleman

Project Summary: Students who participate in a Summer Undergraduate Research Experience in Statistics at Wofford College will receive a broad overview of the research process in the discipline of Statistics. While pursuing original research on one of the two problems presented to them for study, students will be encouraged to recognize the stage of the research process they are currently engaged in. Students will learn the stages of the research process as identifying an open problem and conducting a literature review; choosing or developing methodology towards solving the problem; implementing methodology that leads to either solving the problem or making progress towards its solution; generating tables, graphs, or figures from findings; summarizing results; considering future work; and disseminating the new knowledge. Students will perform miniature literature reviews; learn relevant programming languages; prepare tables, figures, and data to adequately represent results; prepare discussions about graphics; write abstracts for submission to relevant conferences; and enhance their oral presentation skills by giving routine presentations. When students master these skills, it helps them prepare to pursue other REUs in future summers as well as to consider earnestly graduate study.

Project Title: National Research Experience for Undergraduates Program (NREUP)

Project Director: Mihhail Berezovski

Project Summary: The goal of this NREUP proposal is to provide undergraduates from underrepresented groups with an opportunity to perform data-enabled industrial mathematics research by exposing them to problems outside of academia that are mathematical and data-driven in nature. The overarching learning goal is that the students will have the capability to conduct authentic data-enabled research, based on original real-world problems provided directly by businesses and industry. This experience is expected to enhance students’ overall development and to equip them with data analytics and mathematical modeling abilities, problem solving skills, creative talent, and effective communication necessary for various kinds of employment. This early educational experience will foster students’ interest in data-enabled science, help them to make informed decisions and promote their career development. Students will work in a collaborative effort directly with industrial partners while being mentored by program director.

Project Title: Researching the Structure of Graphs from Finite Incidence Geometry with the Aid of Computer Algebra Programs

Project Director: Wing Hong Tony Wong & Brian Kronenthal

Project Summary: In this project, we concentrate on research problems motivated by finite incidence geometry, with special emphasis on its association with experimental mathematics. Common structures in finite incidence geometry include projective planes, affine planes, and generalized quadrangles. Finite incidence geometry has close relationships with graph theory, combinatorics, abstract algebra, linear algebra, mathematical games, and many other fields. It is deep within the realm of proof-based theoretical mathematics, but to understand finite geometric structures, it is often useful to experiment with concrete examples. The size of these structures grows very quickly, and this is where experimental mathematics and its use of computer algebra programs can play a significant role. Four underrepresented minority students will complete this project over nine weeks during Summer 2020. They will learn how to use computer algebra programs to conduct mathematical investigation, formulate research questions by investigating mathematical structures, finding patterns, and identifying fundamental properties, apply newly-learned knowledge in theoretical mathematics to prove conjectures, and communicate mathematics effectively and confidently in both oral and written form. We anticipate student presentations at mathematical conferences and at least one publication in a peer-reviewed mathematics journal.

Project Title: The Summer Program in Research and Learning (SPIRAL) at American University

Project Directors: Monica Jackson

Project Summary: The Summer Program in Research and Learning (SPIRAL) at American University is a research experience for undergraduates (REU) aimed at providing a mentoring structure for women and underrepresented minorities that promotes active learning and engagement in problems in statistics and mathematics. SPIRAL is an awarding winning REU in existence for over 16 years. The seven-week program will be three-pronged: (1) Students will participate in research seminars in mathematics and statistics, in which research projects will be investigated in teams. Each team will write a final paper discussing their results and give an oral presentation. (2) There will be an intensive four week course – emphasizing computer skills, with problem workshops and daily homework. (3) One day a week will be devoted to professional development and career awareness, enhancing the students’ view of the mathematical world. Development activities include learning to write in mathematical software packages, learning to analyze data using statistical packages, training for the GRE, and visits from researchers in industry, academia, and government to share career opportunities. Finally, all students will attend the 25th Conference for African American Researchers in the Mathematical Sciences (CAARMS25) and present preliminary findings of their research project. In previous years, SPIRAL students won first place in the student poster competition at CAARMS.

Project Title: Central Convergence REU

Project Director: Brandy Wiegers

Project Summary:Central Convergence REU provides early-career undergraduate students who have at least a year and a half remaining in their degree program at the time of their REU application a unique summer research experience that encourages them to more fully immerse in the mathematical professional community. The summer project begins with a growth mindset-focused training in problem-solving to prepare them to approach research problems. Then students engage in the research for seven weeks while also receiving professional development mentorship to provide these students the tools and knowledge to engage more in the mathematics profession. By the end of the REU, students have contributed to authentic mathematical research, worked with computational mathematical tools, and practiced multiple forms of mathematical communication. After the students leave the experience they are prepared, encouraged, and supported to find future experiences (research, conferences, courses) to further participate in the professional field with coordinated post-REU activities. The 2018 Central Convergence REU was a success with five-MAA funded students engaging in research that they presented at SACNAS with the support of MAA NREUP and SACNAS traveling funding.

Project Title:Computing Change Using Partial Differential Equations

Project Director: Nessy Tania

Project Summary:Our summer research program will be focused on analysis and computations of partial differential equation (PDE) models. We will begin by introducing students to common PDEs and how to solve them using Fourier series or numerical approximations. There will be two research subgroups: one studying instability in the Euler equation in fluid dynamics and another focusing on pattern formation in a spatial population model. Students will use and develop tools from theoretical mathematics (functional analysis and dynamical systems) to applied mathematics (modeling and numerical approximations). In addition to learning mathematics outside our standard available courses, our undergraduate researchers will build their capacities for mathematical exposition, both in writing and in oral presentations. To complement the mathematical content, students will interview minority professional mathematicians and lead discussions on issues surrounding their URM status. The summer program will end with attendance at MathFest, where students will present their work and engage with the broader mathematical community. We will measure the effectiveness of our program by tracking whether students (particularly those in their first/second years) take further mathematics courses and by measuring their retention within the mathematical sciences post graduation.

Project Title: CMAT - Computational Mathematics at Tarleton

Project Director: Tom Faulkenberry

Project Summary: The proposed NREUP project is called Computational Mathematics at Tarleton (CMAT). With this collaborative, cross-disciplinary project, we aim to stimulate intellectual curiosity and develop transferrable mathematics 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 a fully immersive 8-week research experience, where students will complete collaborative research in computational mathematics, with specific projects in mathematical modeling of cognition and the dynamics of billiard systems. The results of this research will not only contribute to the body of scientific knowledge in these fields, but also contribute to the development of these students' knowledge and research skills related to mathematics and computational science, inspiring these students to persist to graduation, pursue further STEM-related educational opportunities, and ultimately seek careers in the mathematical sciences.

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

Project Director: Hyunju Oh

Project Summary:Students will be introduced to the fundamental concepts of Game theory and Coding theory. In game theory and applications, students will learn Nash equilibria and evolutionarily stable strategy and teach them how to use computational tools (NetLogo), as well as analytical tools (optimization 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, students will learn correcting and detecting errors, information rate, and distance of codes. Then we will show some examples of codes such as linear and cyclic codes, Hamming codes and BCH codes. While learning coding theory, students will review the basic concepts of linear algebra and will learn some concepts of abstract algebra such as finite fields and polynomials which are the basis for understanding the cyclic codes. Participants will have a choice of two projects 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.

Project Title: Long-time Behavior and Allee Effects in Predator-Prey Models

Project Directors: Huiqing Zhu

Project Summary: The Allee effect was shown to bring essential changes to population dynamics and it has drawn considerable attention in almost every aspect of ecology. In our previous MAA-NREUP project that have been completed in summer 2018, the influences of hunting cooperations and Allee effects in several predator-prey models have been investigated. We plan to lead our undergraduate researchers to continue our research program in summer 2019 since our students were well motivated and they made some significant progress in those topics. One of our main motivations is to study the synergistic effects of Allee effects in prey for predator-prey models since there is not much work having been done in this area. We will also investigate the impact of Allee effects to the Turing pattern formations due to the spatial interactions in the three species predator-prey diffusive model. The proposed research project will have a positive impact on participants’ future academic achievements.

Project Summary: Building up on the successful summer 2018 MAA NREUP at UTRGV, we will continue working with our students on stochastic partial differential equations. In particular, we will study exact and numerical solutions of stochastic Schrodinger equations with variable coefficients. We will also study the existence of solutions of systems of stochastic Burgers equations using graph theory. Students will receive extensive training on partial differential equations, probability theory, stochastic processes and analysis and their numerical solutions/simulation using matlab. Students are expected to write up a manuscript of the results coming out of their work during the event and submit it to a peer-reviewed journal. They are also expected to present their research orally and through posters in national conferences.

Project Title: Vaccination Games

Project Directors: Ajanta Roy

Project Summary: The research project we will offer to students belongs to the field of mathematical biology and can be informally called "vaccination games.'' It involves applying game-theoretic methods to individual decisions on the use of personal protective measures against an infectious disease. In particular, it addresses the following fundamental question: Can an infectious disease be eradicated through voluntary participation in personal protective measures, such as vaccination? Individual-level vaccination (or other personal protective measure) decisions can be modeled by game theory since the strategy of a given individual depends on what the rest of the population does: if a sufficient proportion of the population is already immune, then even the slight cost or risks associated with vaccination can outweigh the risk of infection. Students will pick an infectious disease to study. They will use an existing epidemiological model of the disease to obtain the information which is going to be fed into the game-theoretic model of voluntary vaccination decisions for that disease. The students will solve the game-theoretic model and identify optimal vaccination strategies.

Project Title: Central Convergence REU

Project Director: Brandy Wiegers

Project Summary:Central Convergence REU will provide early-career undergraduate students, who have at least a year and a half remaining in their degree program at the time of their REU application, a unique summer research experience that will encourage them to fully immerse in the mathematical professional community. The summer project will begin with a growth mindset-focused training in Mathematica and problem-solving activities to prepare them to approach research problems. Then students will be involved in their research projects in epidemiology modelling and mathematical biology for five weeks, while also receiving professional development mentorship to provide them with the tools and knowledge to be more engaged in the mathematics profession. By the end of the REU, students will have had an authentic mathematical research experience, worked with computational mathematical tools, and practice with multiple forms of mathematical communication including talks, posters, and written articles. Through the coordinated post-REU activities, students will leave the experience being prepared, encouraged, and supported to find future experiences such as other research opportunities, conferences, and related courses as they pursue their careers in mathematics.

Project Title:Howard's NREUP

Project Director: Dennis Davenport

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 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. Students will be given continued group support and valuable role models after the completion of the seven-week program. As a follow-up, each student will be required to enroll in a 3-credit hour readings course in the Fall of 2017 to continue their research and to learn more about combinatorics. Another goal is to have academic year undergraduate research embedded into the mathematics curriculum at HU. This project will be used as a proof of concept on how to better use undergraduate research at HU during the academic year. Hence, an important component will be tracking the students once they complete the program.

Project Title: Summer Program in Research and Learning (SPIRAL)

Project Director: Leon Woodson

Project Summary:The primary goal of The Summer Program In Research And Learning (SPIRAL) program is to provide a mentoring structure for underrepresented minorities and women that promotes active engagement in mathematics and statistics through a Research Experience for Undergraduates (REU) program. With a supportive structure, the participants will be encouraged to remain in math and statistics with the hopes that they strengthen the diversity of the talent pool of fully trained mathematicians and statisticians in academia, government, and industry. The 7-week program will be three-pronged: (1) Students will participate in research seminars in math/stat, in which research projects will be investigated in teams. The student will meet with their research team Monday-Friday. Students will participate in scientific lectures in the mornings. In the afternoon, the students will work on research project that relate to the material learned from the morning. Each team writes a research paper and gives an oral presentation. (2) There will be an intensive 5 -week course - emphasizing proof, with problem workshops and daily homework- consisting of three modules, foundations of mathematics, applied mathematics and statistics that connect to the research projects. Graduate students assist the research mentors with the students’ research. (3) One day a week will be devoted to professional development with visits from researchers in industry, academia, and government to share career opportunities.

Project Title: Graph Theory and Percolation

Project Director: Lazaros Gallos

Project Summary:The MAA/NREUP students will be part of the larger DIMACS REU cohort and will be mentored by Rutgers faculty. In this project the students will explore the core concepts of graph theory and percolation. The projects will illustrate the practical potential of mathematical models in applied problems, such as information spreading in social networks or disease transmission. The projects aim to the development of the students mathematical intuition and the improvement of their analytical and modeling skills. The students will form research teams, joined by additional students in the DIMACS REU program. One team will study aspects of multi-layer graph theory in connection to percolation. The other team will study the frog model and focus on modifying various aspects of the model, inspired by possible applications. We will expose the students to a broader range of possible projects, trying to increase their interest to diverse research topics and research areas. Students will participate in planned social and professional activities such as picnics, cultural day, weekly seminars, workshops, and field trips giving the students more opportunities to interact socially. Students will be introduced to industrial research by a trip to DIMACS partner industry (IBM). The students will be given a tour of the facilities and attend several technical presentations. Students will be encouraged to take advantage of DIMACS activities including the many on-going seminars and workshops.

Project Title: Experimenting with Mathematica and Magma

Project Directors: Mehmet Celik

Project Summary:In this summer research program students will look at the following two problems: Conformal maps: If a map preserves the shape of small-scale features, it is conformal. Many problems in the fields of engineering, physics, and mathematics are formulated in regions with “inconvenient geometries”. By a suitable conformal map, one may transform the problem from an inconvenient region into a nice one and after solving the problem by using the same conformal map one can transform the solutions back to its original region. Participants will investigate how smoothness of a conformal map relates to smoothness of a boundary point of a region by using the computer algebra system Mathematica. Linear Complementary Dual Codes from Strongly Regular Graphs: Error-correcting codes are necessary to detect and correct errors in communication. As we increasingly rely on intelligent devices, our automated homes and medical devices are targeted by intruders using side-channel and fault noninvasive attacks. Finding new classes of error correcting codes that can withstand these attacks is crucial to security. Participants will explore codes from adjacency and incidence matrices of strongly regular graphs by using the computer algebra system Magma. Participants will do research by testing with computer algebra systems. The group work will help organizers to maintain communication among participants, set a collaborative learning atmosphere, foster a sense of community, and promote exchange of ideas

Project Title:Game Theory and Applications

Project Directors: Hyunju Oh

Project Summary: Students will be introduced to the fundamental game-theoretical concepts such as Nash equilibria and evolutionarily stable strategy and taught how to use computational tools (NetLogo), as well as analytical tools (optimization 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. The students will further 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.

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Project Title:Allee Effeccts and Chaos in a Food Chain Model

Project Directors: Zhifu Xie

Project Summary:In population dynamics, the spatial distribution of populations can be affected by depressed growth rate at lower density, which is called the Allee effect. The Allee effect was shown to bring essential changes to the population dynamics and it has drawn considerable attention in almost every aspect of ecology. However, there is not much work that has been done for food chain or food web models with an Allee effect in prey. An interesting question is how the dynamical properties of a food chain model can change when the prey experiences Allee effects. We propose to investigate the dynamical complexity of the ODE and PDE version with Allee effects in prey. Undergraduate researchers will be asked to investigate numerically and theoretically the possible rich dynamical properties of the ODE version. They will learn how to conduct linear stability analysis on equilibrium solutions to understand the long-time behavior of the predators and prey in an ODE system. Several workshops will be held for them regarding (A) how to read research paper, (B) how to write a research paper and use LaTex, (C) how to prepare for a poster or oral presentation, and (D) how to apply to graduate school. Students are expected to submit a final report in a research paper format and give a presentation in local or national conferences. Students may also continue to work with mentors on the PDE version of the problem in the Fall semester of 2017.

Project Title: Stochastic Burgers Equations and Extensions

Project Directors: Tamer Oraby

Project Summary:We are planning to host an REU event. It will involve training four undergraduate students on two problems with the promise of submitting the results to peer-reviewed journals. The research problems that we are proposing for this REU are challenging but certainly doable within the 8 week time frame we have outlined, particularly with the support of the short courses as an intense review of differential equations (which they have all studied either currently or in a previous semester) and focus on the specific areas they will need to apply to our research questions. We have taught these courses many times before and are confident that we will be able to tailor the course content to our individual needs and time constraints. We are also enthusiastic about structuring in time for practice and application of mathematics modeling programs like MATLAB and Mathematica; these skills will serve our students well through their last years as undergraduates, throughout their graduate studies, and beyond.

Project Title: Enhancing and further developing emerging non-smooth optimization methods

Project Director: Milagros Loreto

Project Summary:During the NREUP at University of Washington Bothell (UWB), the students will be introduced to fundamental concepts in nonsmooth optimization such as subdifferentials, optimality conditions and step lengths. Spectral Sampling Gradient (SpecSampling) and Modified Spectral Projected Subgradient (MSPS) are emerging nonsmooth optimization methods. Students will work in one of these two projects: 1) Enhancing and further developing the SpecSampling, where they will learn about the SpecSampling and develop a suitable nonmonotone linesearch technique; 2) Correcting zigzagging of kind II for the MSPS, where they will learn about MSPS and develop heuristics to correct zigzagging behavior of it. They will learn how to use computational tools such as MATLAB to implement the artifacts described above, and to conduce a broad numerical experimentation. The NREUP at UWB will provide an integral program to prepare students on researching, writing and reporting about that research using LATEX, communicating and presenting results, deciding about their own future as researchers and understanding the impact of diversity and inclusion on their work. The PI expects to submit at least two papers to referee-based journals, and to present this work at local conferences and the next Joint Mathematics Meeting.

Project Title: Modeling Cognitive Structures and Dynamics on Topological Domains

Project Directors: Brett Sims

Project Summary: This research on modeling cognitive and social dynamics will expand on conjectures developed during the 2016 NREUP at BMCC. In 2016, students developed a homomorphism that maps over interpretations to define the “gluing” of two philosophical substructures, they also described coordinated social action, via homotopy lifting. In this work students will apply theory of a simplicial complex with attention to affine groups, G-sets, and deck transformations to describe and analyze autonomous and externally stimulated cognitive dynamics. In addition, students will explore applications in homology theory, with the objective of developing a consistent definition for an “orientable or non-orientable” psychological substructure. Analysis will include the effects of specific types of philosophical “gluing” on individual or social “orientation”. Students will define qualitative psychological substructures (philosophical, attitudinal, motivational) and their relations based on existing empirical research. The “total” psychological space will be treated as an n-sorted topological structure under such theory as model theory, theory of a simplicial complex, and group theory. Topological representation of the, generally non-metricizable, psychological space permits a mathematical characterization of qualitative cognitive dynamics that can be coordinated with the metricizable physical spatio-temperal (behavioral) space.

Project Title: Game Theory and Applications

Project Directors: Hyunju Oh

Project Summary: Students will be introduced to the fundamental game-theoretical concepts (Nash equilibria and evolutionarily stable strategy) and taught how to use computational tools, as well as analytical tools to identify such strategies in real game theoretical models with applications in biology or medicine (cat vaccination to prevent Toxoplasmosis or vaccination to prevent Rift Valley Fever). The students will further be trained in all aspects of research, starting with the ethics code, going through 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.

Project Title: Modeling Social Aggregation on Topological Domains

Project Director: Brett Sims

Project Summary: The mathematical study of community formation as a social aggregate and its dynamics has been an interest among social scientists, and in particular Home Land Security where the formation of a social aggregate can be a threat based on the philosophy, attitude, and resources of the aggregate. Beyond national security, an economic question of interest is: to determine the probability that a particular social aggregate will be formed having a desired spending characteristic? This research on social aggregation will build on conjectures developed during the 2015 NREUP at BMCC. Significant changes in a community’s philosophical structure influence aggregation dynamics. Modeling philosophical systems via topological/simplicial structures, students found that a “glued” philosophical structure has the universal property in the sense of category theory. In this work students explore application of interpretation theory on free modules to give a formal mathematical treatment of compatibility issues involved with “gluing” philosophies. This research also aims to explore the characterization of the individual as a generalized operator with application to agenda or issue propagation “speed” through-out a community, during person-to-person communication.

Project Title: Data Dimension Reduction Techniques and its Applications

Project Director: Kumer Pial Das

Project Summary: The MAA/NREUP program at Lamar University is designed to allow 5 talented undergraduate students the opportunity of working on research involving data dimension reduction techniques. Dimension reduction - using matrix factorizations - can be used to ascertain the underlying structure of the data and to find correlations among the data attributes. Interest is increasing in the study of singular value decomposition (SVD) and non-negative matrix factorization (NMF) due to the many applications in various research areas of mathematical sciences, and computer science. During the six weeks students will engage in research involving a comparatively recently developed matrix factorization technique known as Nonnegative Matrix Factorization (NMF). Students will be introduced to the mathematics behind NMF. Low rank approximation is a special case of matrix nearness problem. When only a rank constraint is imposed, the optimal approximation with respect to Frobenius norm can be obtained from the NMF. The low rank matrix will give a cleaner, more efficient representation of the relationship between the data elements. NREUP participants will investigate various challenging matrix factorization problems of current interest related to big data analytics such as data dimension reduction, pattern recognition, and image processing.

Project Title: Identifying Sieves and Primitive Integer Triples Using the OEIS

Project Directors: Eugene Fiorini & Byungchul Cha

Project Summary: This project we will concentrate on research problems associated with the On-Line Encyclopedia of Integer Sequences® (OEIS®) and its role in stimulating new research. Sequences play an important role in number theory, combinatorics and discrete mathematics, among many other fields. They enumerate objects in sets and define relationships among items or properties shared between them. Integer sequences have inspired mathematicians for centuries. The quest to compute new, larger terms in important infinite sequences is harnessing the power of computing and promoting the use of new paradigms in distributed and cloud computing as well as Big Data. A particular emphasis of our project will be to discuss the important role the OEIS has played in developing conjectures in areas that include number theory, algorithmic and enumerative combinatorics, combinatorial number theory, and many other mathematical fields, as well as the tools necessary for identifying such conjectures. Students from San Diego City College and Northeastern University will join other students participating in the Muhlenberg College REU program to form research teams to work on each problem. One research team will concentrate on generalizing algorithmic procedures for generating all primitive triples satisfying quadratic homogeneous equations in . Another team will look at several general questions of a subclass of sieves and develop sieve-like algorithms for producing sequences.

Project Title: Mathematical Modeling of Neglected Tropic Diseases

Project Director: Apillya Lanz

Project Summary: The NSU NREUP Program at Arizona State University (ASU) will follow the structure of the successful Mathematical and Theoretical Biology Institute (MTBI) which is also held at ASU. Four NSU students will be participating. They will identify their own research projects on modeling the transmission dynamics of neglected tropical disease (as identified by the World Health Organization). Under the guidance of their mentor, students will learn how to develop mathematical models via identifying the problem, determining the necessary assumptions, finding the interrelationships among the variables, constructing a model, and interpreting the results given by their models. While working on their projects, students will learn how to modify a simple compartmental model such as SIR and determine the system of differential equations that represent dynamics of a disease and how to interpret the solutions obtained for the problem.

Project Title: Improvements in the Distributional Theory of the Multiple Window Scan Statistic for Unusual Cluster Detection with Application to Sports Streaks

Project Director: Deidra Coleman

Project Summary: Students who participate in this research will work on one of two research projects focused on expanding the theory related to and applications of the multiple window scan statistic for unusual cluster detection. One project is in the direction of improving the computational time of the algorithm for computing the exact distribution of the statistic with application to success clusters within the performance of Professional Bowlers Association (PBA) players. The other project is towards adding to the body of literature relevant knowledge of computing times with application to failure clusters within the performance of National Basketball Association (NBA) players. This work is important for understanding the phenomenon known as the "hot hand" or the notion of being "cold." The "hot hand" is known as a period of time when an athlete may be performing usually well, that is, when an athlete is expected to be successful at nearly every attempt at goal. The goal may be a successful shot in a basketball, a hit at bat in baseball, or a strike in bowling. An athlete is "cold" when he or she is expected to fail at nearly every attempt at goal. Students will learn to perform miniature literature reviews; learn Fortran and R; learn LaTex; extract secondary data; prepare tables, figures, and data to adequately represent results; write up discussions about graphics; and prepare summaries, conclusions, and future work.

Project Title: Graph Theory Applications in Epdiemiology

Project Summary: The MAA/NREUP program at DIMACS in Rutgers is a basic component of a broader REU program that allows the students to participate in the summer research experience as members of a larger interactive group. This year the MAA/NREUP students will work on the following problems: Computational Ramsey Theory: Ramsey theory studies the existence of unavoidable patterns. A mathematical object is colored - its components are each assigned a particular "color" from a finite set of colors. The question is whether a particular type of structure exists within one of the colored subsets. This project will explore the Rado numbers for other equations, and in some cases with more than two colors. The methods may entail theoretical methods related to the number theory of Diophantine equations, but will also include work on the computational and algorithmic methods used to compute Rado numbers efficiently. Mathematical modeling of pathogenic transmission: The students will develop and study a modified water-borne pathogen model for cholera outbreak. They will particularly address the following questions: What is the vaccination threshold or minimum vaccination coverage based on the basic reproductive number for a cholera outbreak, when the disease is seasonal and non-seasonal? What will be the long term behavior of the solution for the susceptible population? What are the conditions and the asymptotic form for the disease-induced death population? This quantity is useful for the estimation of the probability to study population’s long term connectivity and its ability to survive against disease invasion. During this research project the students will explore core concepts of dynamical systems, population heterogeneity, sensitivity analysis, and experience the rigorous process of reproving some of the well known threshold theorems.

Project Title: Decomposing a Function into Symmetric Pieces: Fourier Series and Self-Similarity

Project Directors: Hyeijin Kim & Yunus Zeytuncu

Project Summary: General idea of decomposing a generic function (or a signal) into a superposition of symmetric pieces is a powerful tool in mathematics and science. A few specific areas it is used frequently include differential equations, signal processing, image coding and information theory. Mathematicians use this tool to solve differential equations, engineers use it to reconstruct unknown parts of a signal from observed parts, and applied mathematicians use it to help researchers to interpolate between parts of various models, such as financial, medical, and biological. In the proposed summer program, we plan to present this general idea using two classical tools of decomposition. Fourier series and spherical harmonics, with applications in different areas of science.

Project Title: Low Dimensional Topology and Topologicial Domains

Project Directors: Noureen Khan & Byungik Kahng

Project Summary: The featured new addition to this year's project is an outreach component. All participants will make a presentation to UNT Dallas Girls, a STEM mentorship/outreach project for middle school girls. The 3D-printing technology will also be included in this outreach event. This year's project aims to continue on the following two problems: Virtual Rational Tangles: The theory of rational tangles was discovered by John Conway, during his work on enumeration and classification of knots. A rational tangle is the result of consecutive twists on the neighboring of a classical link. So, a natural trend would be finding the generalization of rational tangles and classification of virtual links based on virtual rational tangles. Invertible Piecewise Isometric Dynamics in Compact Orientable Surfaces: The dynamics of piecewise isometric systems had been studied primarily in the Euclidean plane. This project aims go beyond the traditional domain and study the dynamics in compact orientable surfaces, both analytically and numerically. This generalization is partly inspired by a granular mixing model in a tumbler,

Project Title: Explorations in Graph Theory

Project Director: Dewey Taylor

Project Summary: Students will participate in a challenging, six-week graph theory experience focused on graph products. Projects involving graph products will be chosen from three areas of graph theory; domination theory, graph coloring and graph pebbling. Students will learn how to read mathematical literature, give mathematical presentations, use mathematical software (Sage, Mathematica, Maple) and prepare documents in LaTex.

Project Title: Applications of Complex Analysis

Project Director: Ajanta Roy

Project Summary: This project explores the crucial concepts of geometric properties of complex functions and their applications. Complex Analysis has a multitude of real-world applications to engineering, physics, and applied mathematics. Students will learn necessary background from differential geometry where they will be using ideas and techniques of calculus to apply to geometric shapes to introduce minimal surface. Then we will be using complex analysis to present a nice way to describe minimal surfaces and to relate the geometry of the surface with this description. Students will also explore the geometric properties of harmonic univalent functions and see some of the bizarre behaviors they have that form the basis of an exciting new area of research. They will also investigate some properties and explore new research questions using java applet.

Project Summary: In this research program students will use a novel approach to modeling social aggregation. Students will define qualitative variables, denoting philosophical or theological types and psychological characteristics such as attitude, that will be treated under homotopy and simplicial complex theory to identify relevant domains of individuals that are likely to form social aggregates (communities). Students will also model the spread of group acceptance of or reluctance to an agenda based on group attitude dynamics. The student research participants will work in teams focused on modeling social aggregation formation and evolution under philosophical or theological and psychological motives.

Project Title: Controllability of Classes of Graphs

Project Director: Cesar Aguilar

Project Summary: Networks are increasingly becoming a useful tool to model complex dynamic behavior in science and engineering. An important problem in dynamic networks is the state transfer problem wherein it is desired to transfer the state of a dynamic network to a pre-selected final state. In this REU program, four CSUB mathematics majors will use algebraic graph theory to determine graph structures that inhibit state transfer in networked dynamical systems. The research will focus on understanding the relationship between completely uncontrollable networks and the spectral properties of the Laplacian matrix of the network, and on the relationship between vertex set partitions and network controllability.

Project Title: Radio Labelling of Graphs

Project Director: Min-Lin Lo

Project Summary: In 2001, Chartrand, Erwin, Zhang, and Harary were motivated by regulations for channel assignments of FM radio stations to introduce radio labeling of graphs. A radio labeling of a connected graph G is a function ƒ (think of it as a channel assignment) from the vertices, V(G), of G to the natural numbers such that for any two distinct vertices u and v of G: (Distance of u and v)+|ƒ(u)−ƒ(v)|≥1+ (maximum distance over all pairs of vertices of G). The radio number for G, rn(G), is the minimum span of a radio labeling for G. Finding the radio number for a graph is an interesting, yet challenging, task. So far, the value is known only for very limited families of graphs. The objective of this project is to investigate the radio number of different types of graphs. We will attempt to extend the study to categories of graphs whose radio numbers are not yet known.

Project Title: Applications of Mathematical Biology in Social and Health Services

Project Director: Apillya Lanz

Project Summary: The NSU NREUP program will follow the structure of the successful Mathematical and Theoretical Biology Institute (MTBI) at Arizona State University (ASU). Four students will identify their own research projects on the applications of mathematical biology in social and health sciences. Under the guidance of their mentor, students will learn how to develop mathematical models via identifying the problem, determining the necessary assumptions, finding the interrelationships among the variables, constructing a model, and interpreting the results given by their models. While working on their projects, students will learn how to modify a simple compartmental model such as SIR and determine the system of differential equations that represent dynamics of a disease and how to interpret the solutions obtained for the problem.

Project Title: Mathematical Economics and Finance

Project Director: Wayne Tarrant

Project Summary: In this program students will work on research problems in the general area of mathematical economics and financial mathematics. They will learn about event studies and risk measures in the first two weeks and then apply their knowledge to problems that interest them. They will be given direction on good sources of problems and latitude to choose the question they wish to pursue, also instilling ownership of the work. Students will be taught LaTeX so that they can prepare papers for publication. They will be expected to both present their work at professional meetings and publish their findings.

Project Title:Game Theory and Applications

Project Title: Topologically Equivalent Graphs and Pattern Recognition

Project Summary: This project we will explore the core concepts of graph theory, algorithms, dynamical systems, and their applications to illustrate to undergraduates from underrepresented and economically disadvantaged groups the practical potential of mathematical algorithms and models as they apply to pattern recognition techniques. Pattern recognition procedures developed in this project will be applied to forensic analysis and disease modeling. Pattern recognition techniques have previously been successfully applied to the problem of image processing, as well as using clustering and classification algorithms to provide information about links within such fields as drug profiling. More recently, graph theoretic methods have proven useful in modeling time-intervals of real-case palaeontological data and identifying cutting agents in heron seizures. Developing efficient algorithms and models will advance current research in pattern recognition procedures within both forensic science and disease modeling. Students from Rutgers University, San Diego City College and New York City College of Technology will form research teams, joined by additional students participating in the DIMACS REU program, to work on each problem. One research team will develop algorithms to identify topologically equivalent graphs from given families of graphs with homotopic properties. These algorithms will be used within forensic science to develop and test new pattern recognition methodologies. The other research team will modify the SIWR (Susceptible–Infectious–Waterborne Pathogen Concentration–Recovered) model proposed by Tien and Earn to calculate the threshold associated with the control reproduction number. Throughout the summer students will participate in planned group activities. Upon their arrival, students have a full day of orientation activities followed by an opening banquet with presentations by program leaders. These group activities are part of the overall DIMACS REU program and include social and professional activities such as picnics, “cultural” day, weekly seminars, workshops, and field trips giving the students more opportunities to interact, serving as an interface between social and research activities. Students will be introduced to industrial research by making a trip to a well-known DIMACS’ partner industry or research institute. The students will be given a tour of the facilities and attended several technical presentations. Besides REU activities, students will be encouraged to take advantage of all of the DIMACS activities including the many on-going seminars and workshops..

Project Title: Hurricane Evacuation/Patrolling the US Border

Project Directors: Jan Rychtar, Hyunju Oh, & Joon-Yeoul Oh

Project Summary: Students will be introduced to the fundamental game-theoretical concepts (Nash equilibrium and evolutionarily stable strategy) and taught how to use computational and analytical tools to identify such strategies in models with applications to hurricane evacuation and border patrolling. The students will be trained in all aspects of research, starting with the ethics code, going through 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. Hurricane Evacuation During the hurricane season, residents in southeast coast area experience frequent warnings for hurricanes. The residents need to be evacuated to safety at least 20 to 50 miles away from the impacted area. With a mass evacuation, even 24-hour notice may not be enough since necessities such as lodging are limited and the actual evacuation distance can easily be more than 100 miles. When a hurricane is approaching, the residents prepare with installing blocks on windows, buying gas/food and deciding if and when to evacuate. In general, if they are getting ready too early, the cost to prepare is too high due to the frequent false warnings (long term hurricane path predictions are not yet reliable). However, if the residents wait almost till the end, their lives get threatened (short term predictions are relatively accurate). Moreover, when everybody evacuates at the same time, there will be logistical issues such as traffic congestions and no fuel in gas stations. The goal of this project is to find an optimal evacuation time. The optimal time depends on individual circumstances and risks (for example, a family with young children is in a different situation than a single healthy young person) and the objective is to find the time as a function of the individual risk and the rick distribution within the population. Patrolling the US Border U.S. Customs and Border Protection (CBP), is responsible for securing the border between U.S. ports of entry and has divided the 2,000-mile U.S. border with Mexico among nine Border Patrol sectors. CBP reported spending about $3 billion to support Border Patrol's efforts on the southwest border in fiscal year 2010 alone, and apprehending over 445,000 illegal entries and seizing over 2.4 million pounds of marijuana. The number of border patrol officers has been increased but because of the limitation of patrolling personnel and budget, it is critical to allocate resources appropriately. The goal of this project is to optimize border patrol routes. The infiltrators' goal is to enter US successfully while patrols intend to capture infiltrators to prevent illegal cross-border activities. Students will develop various models with the objective to find optimal routes and patrolling patterns for the optimal border protection.

Project Title: Decomposing a Function into Symmetric Pieces: Fourier Series and Self-Similarity

Project Directors: Hyeijin Kim & Yunus Zeytuncu

Project Summary: General idea of decomposing a generic function (or a signal) into a superposition of symmetric pieces is a powerful tool in mathematics and science. A few specific areas it is used frequently include differential equations, signal processing, image coding and information theory. In the proposed summer program, we plan to present this general idea using two classical tools of decomposition, Fourier series and refinable functions, with applications in different areas of science. Many students learn these or similar tools in different courses but a coherent explanation with real world applications may get lost in the exposition. Our main goal is to get students familiar with this ubiquitous idea and prepare them to apply it to different problems. In the program, students will obtain an in-depth understanding of Fourier series and refinable functions. They will learn about mathematical issues arising in the process of superposing pieces, like different convergence and divergence problems. They will also practice how to overcome these issues using new techniques. We believe this program will provide a valuable opportunity for students to learn more about how to use tools of mathematical analysis to resolve issues arising in real world applications. In addition, students will practice on scientific writing, oral presentations, and team work.

Project Title: Bounded Invariants and Piecewise Isometries

Project Directors: Noureen Khan & Byungik Kahng

Project Summary: The project is a continuation of the investigators' NREUP 2014 project. This year's NREUP project aims to improve those of the previous years' still further, by expanding its scope on the targeted students. Problem 1. Bounded Invariants of Pseudo-Prime Virtual Knots. In [1], M. Hirasawa, N. Kamda and S. Kamda introduced bridge presentations of pseudo-prime virtual knots with real crossings number less than 5, up to mirror images and knot orientations were presented by the knot codes. However, the authors conjectured five open ended questions in Section 6, page 892. We intend to address those problems and extend the table of virtual knots with real crossing numbers less than 5 by including the virtual ascending degree and virtual unknotting number. Problem 2. Trichotomy of Singularities of Bounded Invertible Planar Piecewise Isometries. The dynamics of planar piecewise isometric systems is the best example that visualizes the intricacy caused by singularity alone, with no in influence from non-linearity. One interesting aspect of the singularity is the pseudo-chaotic dynamics it generates, and the characterization of Devaney-chaos was done by the second investigator [2]. However, the full classification of the singularities in relation to the afore-mentioned characterization still remains incomplete. This project aims to address this issue and make further progress toward the complete trichotomy of the singularities. References [1] M. Hirasawa, N. Kamada, S. Kamada, Bridge Presentation of Virtual Knots, Journal of Knot Theory and its Rami cations, 20(6) (2011) 881-893. [2] B. Kahng, M. Cuadros and J. Sullivan, Sliding Singularities of Bounded Invertible Planar Piecewise Isometric Dynamics, International Journal of Mathematical Models and Methods, 8, (2014), 59-64.

Project Title: Dynamics of Evolution Equations Modeling Wave Phenomena

Project Director: Erwin Suazo

Project Summary: Wave phenomena appears naturally in biology (gene competition), quantum physics (particles) and theoretical optics (fiber optics), and evolution equations are a great mathematical model for these phenomena. The ideal type of solutions for the latter are traveling wave solutions (they preserve their shape on time). The use of nonlinear ordinary differential equations (ODEs) to find traveling wave solutions has been a standard tool for mathematicians, and they involve mathematically rich areas such as special functions and Fourier analysis. In particular, the Riccati equation (RE) with variable coefficients has been a useful tool in mathematical biology (RE is better known as logistic equation) and in mathematical physics: diffusion and Schrodinger type models. For the last two, a significant number of the explicit solutions that are available in the literature that can be solved in the real line thanks to RE. Further, RE with variable coefficients can be used to find explicit solutions for the celebrated (constant coefficient) nonlinear Schroedinger equation (NLS) that is a standard model of how light propagates inside of a fiber optic with mathematical rich properties such as being integrable. In this REU students will study explicit solutions for a nonlinear Riccati-Ermakov system with selected variable coefficients, and use the solution of the system to construct explicit solutions for a Nonlinear Schroedinger equation with variable coefficients through the construction of transformations to the standard model with constant coefficients. The techniques that the students will use (based on the PI’s previous research) will provide six parameters. It is expected that students will provide an interpretation of the parameters related to the dynamics of the central axis of symmetry of the traveling wave solution. Also, numerical simulations of more general problems will take place where analytical techniques proposed by the PI’s research can’t work.

Project Title: Computational Explorations in Differential Equations

Project Director: Theresa Martines

Project Summary: Participants will have a choice of two projects during the program. The first project studies the visualization of solutions to nonlinear partial differential equations in the presence of multiple wave solutions. These equations are used to model propagation of surface water waves in shallow canals, hydromagnetic waves in cold plasma, ion-acoustic wavesand acoustic waves in crystals. They will verify the solutions and discover how the free parameters in the solution relate to various physical properties such as the speed, amplitude and width of a moving wave. The participants will also consider complicated multi-soliton solutions to the KdV equation. Then the participants will focus on the free parameters and the visualization of solutions. In particular they will investigate the role of these parameters when there is an interaction of multiple waves. The second project studies the use mathematical models to investigate the role of fertilizer runoff due to rainfall, on the algal population dynamics. Existing mathematical model analyses of allelopathic competition utilize a chemostat-type model and center on the toxin production pathways and not the role of external nutrient variation, which typically is assumed constant. The chemostat laboratory device uses three vessels to culture microbial populations in a homogenous environment with external nutrient control. For two competing species and one constant limited nutrient, a well-known system of three ordinary differential equations models both the internal nutrient and population levels within a chemostat. This model forms the basis for our study, which will incorporate allelopathy and periodic nutrient input and explore the role of external nutrient variation on the competition

Project Title: Radio Labeling of Graphs

Project Director: Min-Lin Lo

Project Summary: In 2001, Chartrand, Erwin, Zhang, and Harary were motivated by regulations for channel assignments of FM radio stations to introduce radio labeling of graphs. A radio labeling of a connected graph G is a function ƒ (think of it as a channel assignment) from the vertices, V(G), of G to the natural numbers such that for any two distinct vertices u and v of G: (Distance of u and v)+|ƒ(u)−ƒ(v)|≥1+ (maximum distance over all pairs of vertices of G). The radio number for G, rn(G), is the minimum span of a radio labeling for G. Finding the radio number for a graph is an interesting, yet challenging, task. So far, the value is known only for very limited families of graphs. The objective of this project is to investigate the radio number of different types of graphs. We will attempt to extend the study to categories of graphs whose radio numbers are not yet known.

Project Title: The Summer Program In Research And Learning (SPIRAL)

Project Director: Dennis Davenport

Project Summary: The seven-week program will be three-pronged: (1) Students will participate in research seminars in which research projects are investigated in teams. Each team will write a final paper about their results and give an oral presentation. The research areas will be combinatorics, combinatorial games or using difference equation as a tool to model biological occurrences. (2) There will be an intensive five-week course - emphasizing proof, consisting of three modules, in set theory, combinatorics and basic foundations of mathematics, (3) One day a week will be devoted to career awareness, enhancing the students’ view of the mathematical world.

Project Title: Noisy Competing Dynamics and Its Applications

Project Directors: Mark Iwen, Hyejin Kim, & Tsvetanka Sendova

Project Summary: Mathematical models are powerful tools for understanding and exploring the meaning and features of dynamical systems, and they have been extensively used in various fields such as biology, physics, ecology, finance, economics and many others. Mathematical models are usually categorized into two groups - deterministic and stochastic. Since noise can play a significant role in the dynamics of some systems, stochastic dynamics can sometimes provide additional insight into real world applications. In this project students will model and simulate the competing dynamics of systems in the areas of biology and economics.

Project Title: Interdisciplinary Research in Graph Theory and its Applications in the Sciences

Project Director: Aihua Li

Project Summary: The two main projects are “Study of Randic Connectivity Indices of Graphs” and “Graph Theory Applications in Modeling Evolution of Chagas Disease Insect Vectors”. The program emphasizes an interdisciplinary approach through both theoretical and applied research. It offers the participants opportunities to explore selected graph theory problems raised from chemistry and biology and to experience original mathematics research and scientific applications.

Project Title: Applying Graphs to Twitter and Brain Connectivity

Project Directors: Eugene Fiorini, Urmi Ghosh-Dastidar, & James Abello

Project Summary: This project explores the core concepts of graph theory, algorithms, and their applications to social needs to empower undergraduates from underrepresented and economically disadvantaged groups to become agents of change in their communities. The projects involve comparison techniques of weighted graphs to analyze brain connectivity and applying time-varying graph properties to social media to formulate computational and algorithmic challenges associated with the social and economic needs of the students’ communities. Students from San Diego City College and New York City College of Technology will form research teams, joined by additional students participating in the LSAMP and DIMACS REU program, to work on each problem.

Project Title: Game Theory and Applications

Project Directors: Jan Rychtar, & Hyunju Oh

Project Summary: Our students will be introduced to the fundamental game-theoretical concepts (Nash equilibrium and evolutionarily stable strategy) and taught how to use computational and analytical tools to identify such strategies in models with applications in biology and/or medicine (cat vaccination to prevent Toxoplasmosis infection or bed-net use to prevent malaria). The students will be trained in all aspects of research, starting with the ethics code, going through 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.

Project Title:Invariants in Low Dimensional Topology and Topological Dynamics

Project Directors: Noureen Khan & Byungik Kahng

Project Summary: The project is a continuation of the investigators’ NREUP 2012 and 2013 projects. This year’s NREUP project aims to improve those of the previous years’ still further, by expanding its scope on the targeted students. Problem 1. Invariants of Virtual Knots. There are inﬁnitely many ﬂat virtual diagrams that appear to be irreducible, but so far there is no known technique to prove this conjecture. More speciﬁcally, how can one tell whether a virtual knot is classical? Or, are there non-trivial virtual knots whose connected sum is trivial? The latter question cannot be resolved by classical techniques, but it can be analyzed by using the surface interpretation for virtuals. Problem 2. Controllability and approximate control of the maximal/minimal invariant sets of a class of non-linear control dynamical systems with singular disturbance. Invariant set theory is one of very few reliable tool, under the singular disturbance that prohibits the use of traditional calculus-based tools. The controllability is often the ﬁrst problem that must be resolved. Here, we focus upon the controllability of the optimal cases, the maximal and/or the minimal invariant sets.

Project Title: Three Species Food Chain Models

Project Directors: Dawit Haile & Zhifu Xie

Project Summary: In population dynamics, the predator-prey system has been extensively studied and the analysis of food chains is an active research area in the biomathematical science. In this project, we will focus on simple food chain models that consist of three species where the third species preys on the second one and simultaneously the second species preys on the first one. A predator is a generalist if it can change its food source in the absence of its favorite food that is very common in nature. There are several interesting cases of simple food chain models of three interacting species based on the types of predators. One such case is where there is a generalist predator and a specialist predator. Another is a case where both predators are generalists or where both are specialists. Students will be divided into two groups and each group will investigate different food chain models. The project will combine numerical simulation and theoretical analysis on the existence of all possible solutions. Students will explore the long time behavior of the food chain models with the goal of finding the ranges of the parameters that lead different dynamics such as stable equilibrium, limit cycle, and chaos. Students are expected to conduct linear stability of equilibrium solutions. They also will be asked to conduct a two parameters analysis and to understand the interactions between the self-reproduction of prey and the self-competition of middle predator. A complete parameter graph will be produced to describe the dynamical behaviors under the interactions.