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.