Mathematical modeling research project for a topic you are interested in
For the project, you can develop a novel mathematical model related to your topic or provide extensions or new applications for an existing mathematical model.
Introduction: (longer than 5 pages), due April 23
The introduction should highlight the topic you are going to study, the related backgrounds and the motivation of your study, what have been done on the topic, as well as what have-not been investigated. Work on the topic using various mathematical arguments and simulation approaches
Final project: (longer than 15 pages), due May 20
More information
It will not be sufficient to study an existing model and present already reported results for the model; there must be a novel component to your project. Although, there is a great deal of flexibility in what you include in your paper and in your oral presentation, there are some general things you should definitely consider.
The Outline: Please prepare an outline of your project. The outline should, of course, be in outline form. It should describe what you plan to do for the topic background (including references you’ve found), what you plan to do for the previous mathematical models section (including references), describe what contributions you plan to make and what analysis and or simulations you plan to perform, finally describe how you plan to conclude your project report.
The Project Report: Your research paper (project report) should include:
• a title page with the standard information
• an abstract summarizing the content of your paper
• an introduction to the relevant background information necessary to understand the problem
• an introduction to the mathematical approach that describes the need for mathematical modeling, previous mathematical investigations, etc.
• a section describing YOUR contributions to the mathematical modeling effort
• a section containing ORIGINAL numerical simulations
• a conclusion section in which you give YOUR OPINION of the mathematical models you’ve researched, a summary of how your contributions effect the current understanding, and suggestions for future investigations
Many styles are possible for the body of your paper and although the exact format is left up to you.
The Oral Presentation: You are required to make a 15-minute oral presentation to me (since the class size is much bigger than expected). Your presentation should be a condensed version of your paper. I will expect to you use overhead transparencies, PowerPoint, a poster or some other type of visual aid to help you explain your topic background and mathematical model/analysis. The 15-minute time slot is firm, presentations which take significantly less than 15 minutes will result in a grade reduction. Please practice your talk and give a polished presentation as you are now the expert on your particular topic and must describe and explain its importance. (Note: On time slots that you are not presenting, your attendance is not a must. However, you are welcome to attend the presentation of your classmates and raise questions.)
Two types of main studies:
(a) Focus on Novel Model Development: One possibility is to focus on the development of a model that describes the mechanisms under the topic. In this case, the majority of your effort will be centered around deriving and justifying an appropriate mathematical model and performing preliminary analyses and simulations; or
(b) Extending an Existing Modeling Approach: Many useful models of interest already exist. Another possible way to arrange your project is to provide a review of previous investigations of the problem of interest, and then extend these modeling frameworks in ways which address related questions that have not been fully answered. In this case, most of your effort will be centered around justifying your model extensions, performing analysis of the new equations, and predicting how your suggested modifications change the previously predicted behavior. This type of project different from the one described above in that the focus shifts from model development to model analysis. That is, you will have a template to follow (from the literature) for what type of analysis will be useful and progress should be possible.
Topics: Your choice of topic may involve constructing/extending your own mathematical model of a given situation, describing a well-known model from science or economics, or describing a class of models including at least one specific example. Here are some ideas to get you started:
• A sample paper: “Surprising Dynamics From a Simple Model”, James Walsh, Mathematics Magazine, Vol. 79, No. 5 (2006), 327-339. How do we model the decision-making process? Suppose that you have two tasks that must be completed in a certain amount of time. While you do one job, the other waits in the queue. There are some surprisingly complicated dynamics that can arise from such a simple system.
• Google “Jstor + The American Mathematical Monthly”: The American Mathematical Monthly has lots of articles that are accessible to students in the subject. A good article will have good references. If you are interested in something, track down some references.
Some interesting articles I came across include:
i. “Finding Good Bets in the Lottery, and Why You Shouldn’t Take Them”
ii. “Parking Cars of Different Sizes”
iii. “The Velocity Dependence of Aerodynamic Drag: A Primer for Mathematicians”
iv. “Large Torsional Oscillations of Suspension Bridges Revisited: Fixing an Old Approximation”
v. “What is the Correct Way to Seed a Knockout Tournament?”
vi. “Which Tanks Empty Faster?”
vii. “The Forced Damped Pendulum: Chaos, Complication, and Control”
• More challenging problems: The Consortium for Mathematics and its Applications (COMAP) runs an annual Mathematical Contest in Modeling (MCM) for undergraduate math majors. One option for your project would be to choose one of the MCM questions from previous years and develop a model for the situation described in the problem. The COMAP website (https://www.comap.com/undergraduate/contests/mcm/previous-contests.php) keeps a list of all the old problems stretching back to 1999, so you can browse through the questions and find one that interests you.
• From some other resources