Undergraduate
Poster Session

Abstracts

D. Christopher Arney, Army Research Office
Title: Mathematics Research: Opportunities and needs for the Army in probability and statistics, computational mathematics, discrete mathematics, complex analysis, and cooperative systems
Abstract: "New sciences are emerging. And they measure themselves not by any of today's scientific yardsticks, but by the needs of tomorrow's technologies." (K. Devlin, Goodbye, Descartes) This presentation provides a description of the opportunities for research in the areas of probability and statistics, computational mathematics, discrete mathematics, complex analysis, and cooperative systems to enhance future technologies. This information will help future researchers to understand collaboration with the U.S. Army. Topics covered include the Army's research goals, current thrusts, support programs, and an overview of Army Research Office's (ARO) mathematics programs, which sponsors projects to enhance the decision-making, communication, intelligence, logistics, and combat system performance for future Army systems. In addition, the speaker's innovative research in pursuit-evasion communication, value of information, and the framework for machine (bot) language will be presented as examples of cooperative systems research.

Lora Billings, Montclair State University
Title: The wonderful world of diseases
Abstract: Epidemiology is a field that interests almost everyone. Modeling disease spread can be as easy or complicated as you can handle. In this talk, I will discuss some of the interesting projects that my students have tackled in our quest to understand disease dynamics and design ways to stop them.

Carlos Castillo-Chavez, Arizona State University
Title: Mathematical and Computational Models in Epidemiology: From epidemics to homeland security
Abstract: In this lecture, I will give a personal perspective on the use of mathematical models to study disease dynamics. I will also highlight the applications of epidemiological frameworks in the study of the deliberate release of biological agents and their application to homeland security.

Robert L. Devaney, Boston University
Title: Chaos Games and Fractal Images: Exciting students about mathematics
Abstract: In this talk we will describe some of the beautiful images that arise from the "Chaos Game." We will show how the simple steps of this game produce, when iterated millions of times, the intricate images known as fractals. We will describe some of the applications of this technique used in data compression as well as in Hollywood. We will also challenge students present to "Beat the Professor" at the chaos game and maybe win his computer.

Ryan Girard, Front Range Comm. Coll.
Title: Implementing a Required Online Homework Component in a Traditional Calculus Class - Lessons Learned
Abstract: The math department at Front Range Community College implemented a required, online, homework component in our classroom-based calculus sequence. MyMathLab (MML) began in calculus 1 during the Fall of '07. Surveys were administered to calculus 1 classes during the spring, summer, and fall semesters of '07 to collect pre and post MML results. One-on-one interviews were also conducted with four students. Results of the surveys and the interviews will be presented along with an extensive "Lessons Learned" section.

Rudy Horne, Florida State University
Title: Solitary Waves in Discrete Media in the presence of Four-Wave Mixing Products
Abstract: In this talk, I will discuss solutions that arise in a vector discrete model of the Nonlinear Schrodinger equation where nonlinear inter-component coupling and four-wave mixing are taken into account. We show that the solutions to this model give rise to two single mode branch solutions as well as two mixed mode branch solutions. These solutions are obtained explicitly and their stability is analyzed in the so-called anti-continuum limit. Also, we connect this analysis to recent experiments that motivated this work.

Fern Hunt, National Institute of Standards and Technology
Title: Convergence and Sensitivity of a Multiple Sequence Alignment Algorithm Dynamical Queueing Systems
Abstract: In the last two decades there has been an enormous increase in the number of protein sequences whose biological function must be analyzed. The principal non-experimental method for doing this is a technique known as multiple sequence alignment. Most current methods use dynamic programming based procedures for minimizing the objective function and constructing the alignment. In this presentation we will discuss a method that is based on recasting the alignment problem as the search for an optimal policy in a Markov decision problem. This approach has the advantage of reducing the construction of an alignment to solving a linear programming problem. Secondly the sensitivity of the alignment to perturbations in the cost matrices PAM or BLAST can be assessed. Finally, although the solution optimizes the average or expected cost, one can show under appropriate conditions on the behavior of the underlying random process, that the alignment obtained by this method minimizes the actual cost with probability approaching 1 as the length of the aligned sequence increases.

William Massey, Princeton University
Title: Introduction to Operations Research and Dynamical Queueing Systems
Abstract: Problems may have many different solutions but the goal of a strategic decision is to find the best one. This is the mathematical field of operations research. We initiate the process of model building by applying methods of inference to the problem. Next, we analyze the model and study its properties. Finally, we develop an algorithmic policy from our analysis that leads to the best decision.

The field of queueing theory was invented in the first half of the 20th century to model and design the telephone network. Queueing models are random processes that make significant use of the steady state theory for continuous time Markov chains. In the second half of the 20th century, this theory contributed to the design of the Internet.

This talk discusses the new types of mathematical tools needed to analyze dynamical queueing systems. These methods capture more of the time-varying behavior that would otherwise be washed out by the classical steady state analysis. Moreover, we can develop policies for the strategic design of communication systems and services by using the dynamic optimization techniques of classical mechanics.

James Meiss, University of Colorado, Boulder
Title: Visualizing Dynamics: The Standard Mapping
Abstract: We will explore of many of the phenomena of area-preserving mappings using the Macintosh application "StdMap." Area-preserving mappings provide the simplest model of chaotic behavior in conservative systems. We will show how to find periodic, quasiperiodic and chaotic orbits.

Frank Morgan, Williams College
Title: Surfaces with Density and the Poincaré Conjecture
Abstract: Calculus teachers and physicists have long studied surfaces with density (which you integrate to get mass). There has been a recent surge of interest in the geometry of such surfaces, ranging from probability to Perelman's recent proof of the Poincaré Conjecture to recent advances by undergraduates.

Eric Phipps, Sandia National Lab
Title: A Recent Perspective on Predictive Computational Science: Educational Needs and Opportunities
Abstract: Much of the computational science research and development funded through the Department of Energy and other government funding sources has focused on "Predictive Computational Science". The goal of this work is to replace as much as possible experimental testing with computational simulation in making high-consequence decisions, as testing may be too expensive, impractical, or forbidden. Examples include go/no-go decisions on space shuttle flights, licensing long-term underground storage of nuclear waste, and certifying the nation's nuclear stockpile. In this talk I will give an overview of the processes the national labs are undertaking to attempt to develop predictive computational capabilities, based on the concepts of verification, validation, and uncertainty quantification. I will then provide a recent perspective on the unique educational needs for future computational scientists this research focus requires and some of the opportunities it motivates.

Gareth Roberts, College of the Holy Cross
Title: Using Algebraic Geometry in Celestial Mechanics
Abstract: Many important questions in the Newtonian N-body problem can be formulated as the solution set of a system of polynomial equations in several variables. Examples include finding central configurations, locating equilibria in restricted N-body problems, Saari's conjecture and proving the number of relative equilibria equivalence classes is finite. Unfortunately, the equations governing these problems are often quite complicated, potentially involving thousands of terms and multiple parameters. Modern techniques from algebraic geometry such as Groebner bases, resultants and BKK theory have recently been employed with great success on many of these problems. We will present a few examples, including the work of some talented undergraduate researchers.

Peter Staab, Fitchburg State College
Title: Linear Algebra and Linear Programming in a Finite Mathematics Course
Abstract: For students with weak mathematical skills, understanding many of the basic algorithms in Linear Algebra and Finite Math is difficult, despite its simple nature of adding, and multiplying numbers. The finite mathematics course at Fitchburg State College is designed for students that need to satisfy a math requirement for their degree and many of the students are weak in the basic mathematical skills. I have recently developed software that runs natively in the Firefox browser that allows students to perform basic matrix operations including finding inverses, step-by-by Gaussian Elimination and pivoting as well as the use of the tools in linear programming. The students are required to do a project using the software and write a short paper documenting the problem. In this talk, I present the software that the student use as well as much of the pedagogical reasoning behind the choices made in developing and using the software in the course described above.

Diana Thomas, Montclair State University
Title: Optimization of Student Learning
Abstract: Professors spend a lot of time teaching, preparing to teach, and assessing students. During the past year I spent time evaluating the purpose of each teaching related duty I preformed and revamping it to extract as much as I can from students, beginning with a promising syllabus. In this class I will review two classes I taught after changing my teaching strategies; Calculus I and Linear Algebra. I will discuss:

• How to develop a promising syllabus.
• How to keep continually assess your teaching during the course.
• Focus on specific skills and mastery (such as mastery of chain rule or mastery of proof organization).
• How my students fared in the following semester.
• Things I will change next year.

Abdul-Aziz Yakubu, Howard University
Title: Malaria Model and Optimal Drug Administration Protocols
Abstract: We introduce a deterministic malaria model for determining the optimal drug administration protocol that leads to the smallest first malaria episodes. To explore the effects of administering the malaria drug on different days in the wet season while minimizing the potential harmful effects of drug overdose, we define 55 drug administration protocols. Our simulation results support the clinical studies of Coulibaly et al. at a site in Mali. In addition, we provide protocols that lead to smaller numbers of first malaria episodes than the protocol of Coulibaly et al.