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.

California State University, Bakersfield

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.

California State University, San Bernardino

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.

Norfolk State University

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.

Rose-Hulman Institute of Technology

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.

Rutgers University

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..

University of North Carolina, Greensboro

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.

University of Michigan Dearborn

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.

University of North Texas, Dallas

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.

University of Puerto Rico

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.

University of the Incarnate Word

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