Math 824 (Algebraic Combinatorics)
Fall 2010
Final Project

Project requirements

1. Read a research article on a topic in combinatorics. You can either choose your own article (for suggestions in how to do so, see below) or you can ask me to recommend an article on your favorite topic. In any case, you must get my approval for the article you have chosen; two people cannot report on the same article. You must make your choice by Monday, November 8.
2. Write a short report (approximately 4-5 pages) summarizing the results of your chosen article. The report must be written in good mathematical style (see below). The final report is due on the last day of classes: Wednesday, December 8.
3. Read and make constructive comments on another student's written report. (I will determine randomly who reads whose report.) Your comments are due on the day of the final exam: Friday, December 17.

General instructions on content and style

• Your paper should be in narrative style: the reader should be able to start at the beginning and read to the end. In particular it should not be just a list of theorems or equations. (It is appropriate, however, to state the major results word-for-word.) It should give an overview of the article, emphasizing the general ideas and the relationships among the results in the article. Remember that your audience is a fellow Math 824 student, who has roughly the same background knowledge as you do, but has not read the article.
• You should include at least one proof. This should not be copied word-for-word from the article. To accomplish this, study the proof until you understand it well (you may have to fill in details that the author omitted). Then put the article away and write a proof of the result. You should also illustrate special definitions and results with your own examples (different from those in the article).
• You should also include a bibliography. This will certainly include the article you are studying, probably a general reference work such as Stanley, and possibly books or articles on related subjects.
• You should prepare your report using LaTeX. If you need to include figures, it is acceptable to draw them by hand.
• Some links with good advice about mathematical and technical writing:
  • Writing in the Sciences (UNC Chapel Hill Writing Center)
  • A Guide to Writing Mathematics and Mathematical Writing Checklist (Prof. Kevin Lee, Purdue)

Guidelines on comments

• Comment constructively. Assume that the writer is doing his or her best to convey information effectively. If some part of the report is not achieving that goal, your job is to say so candidly and neutrally.
• Make your comments specific. Compare these two comments:
  1. "I don't understand whether the definition of the adjoint functor of a hypersemisimplex also applies to a semihypersimplex."
  2. "I don't understand Definition 3.14."
  
  Comment 1 is helpful: it tells the author what to do in order to make the paper clearer for the reader. Comment 2 is much less helpful: the author doesn't know whether to simplify the language, or add an example, or change the notation, or something else.

Suggested articles

The two sources I recommend for articles are the arXiv preprint server and the Electronic Journal of Combinatorics. The arXiv is a preprint server which is widely used by mathematicians to distribute papers pre-publication, while EJC is a well-respected peer-reviewed journal. Both are completely open-access. (You may also look in other sources --- for instance, try typing "Tutte polynomial" or "supersolvable arrangement" into MathSciNet --- or ask me to suggest possible papers.) You can browse both the arXiv and EJC websites for recent papers in combinatorics. If you see a word you like in the title, and the paper looks like a reasonable length (say 10 pages, or at most 15), then read the abstract and see if it looks like something you'd like to read. Check it with me before you start (particularly for arXiv posts, which are not moderated or refereed).