# Math 4740: Course Project

The goal of the course project is to give you an opportunity to explore applications of stochastic processes in a field you love (possibly your major). It will also give you a sense of what it's like to be down in the trenches reading and writing scientific literature.

Find a paper in a peer-reviewed journal in your area of interest which uses a stochastic process to model some "real-world'' phenomenon. I will suggest a few classic papers in different fields, but you are welcome to choose your own after running it by me. Write a review giving a summary and critique of the approach used in the paper. Please address the following questions:

• What real-life behavior are the authors trying to study? Why is it important? (Please remember that I am not an expert in your field. You don't have to write a survey article for me, but try and give some idea of what is going on!)
• Describe the mathematical model that the authors propose, in terms that a classmate could understand. This may require further research on your part.
• If the model is similar to something we have studied in class, discuss how some of the things we know about it might be relevant. (For instance, if the article is using a Poisson process to model the times when a neuron fires, then for two disjoint time intervals, the number of firings in the first interval is independent of the number of firings in the second. Is this realistic?)
• What are some aspects of the real-life behavior that the model does a good job of describing?
• In what ways is the model unrealistic? How could the model be improved?
• Are there other models commonly used to describe this behavior? (The article may give useful references.) Why have the authors chosen this one instead?
• Do the authors compare the predictions of their model (from theory or from simulations) with data from real-life observations or experiments? If so, what did they find?

Here are three examples of good projects from last year's class:

Here are some guidelines on writing and submitting your review:

• Please type it using LaTeX (or LyX). Do not use Microsoft Word. (To help you out, here is a sample LaTeX file and the pdf output it generates.)
• This is an individual project. You are welcome to discuss the project with classmates, friends, professors, or others, but the work you submit must be your own. If someone provides useful comments, it would be courteous to thank them in the paper.
• Write as if your work were going to be published in a professional journal. Be sure to reread and revise as needed.
• Concise writing is highly valued in mathematics. Longer is not necessarily better! (A common takedown among mathematicians is "It took you X pages to prove that?") If you can convincingly address all the questions above in 2 pages, I give you props. If you chose a complicated paper and it really takes 15 pages to pick it apart, you also get respect. There is no particular upper or lower bound on the length of your review. Decide what points you want to make and use only the amount of space needed to make those points.
• Give full citations of all sources, including the article you are reviewing. (This means at least: article title, authors' names, journal title and year of publication.) Direct quotes must be indicated as such, and the source named specifically. You may use any citation style common in your field.
• If the article you are reviewing is available online, include a link (use the hyperref package in LaTeX).
• Please print out your project, staple it to the paper you are reviewing and hand it in in class. The first draft is due by 9:05am on Friday, April 18. The first draft should be substantially complete and contain all the required elements. I will return it with comments within one week. The final version is due by 9:05am on Wednesday, May 7. Late submissions (of either draft) will lose value exponentially at a rate of 1% per hour.
• The project is worth 20% of your final grade, broken down as follows:

• A(4%): Clearly explaining the mathematical model proposed in the paper.
• B(4%): Clearly explaining the paper's results. Are they theoretical results (theorems with proofs) or experimental?
• C(4%): Analyzing the strengths and weaknesses of both model and results.
• D(4%): Having a concise, well-organized, professional writing style.
• E(4%): Meaningfully revising and improving on the first draft.
If you have a great idea for something to look at which doesn't quite fit the guidelines of this project, let me know and we can discuss it.