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.