University of Connecticut College of Liberal Arts and Sciences
Department of Mathematics
Tom Roby's Math 3094 Project Page (Fall 2013)
Algebraic Combinatorics (Math Scholars)

Project Information

GENERAL DESCRIPTION: The project consists of two parts:

  1. a typeset paper of 6–10 pages (single spaced); and
  2. a 20-minute presentation to the class, with additional 5 minutes for questions.
You are welcome to work alone or in pairs. In the latter case, I'd expect the paper to be 12–18 pages and the presentation to be about 35 minutes. One advantage to this approach is that it allows you to work deeper or more broadly in certain topics without having to define as many notions from scratch.

PAPER: The paper should be clearly written and comprehensible by the other students in the class. The final version will be due the last day of class; a good first draft is due on 12 November 2013. I will read this and suggest revisions for you to implement in the final version. The paper must be typeset, preferably in LaTeX (or plain TeX if you're a real hacker).

A good paper will include:

  1. motivating background information,
  2. clear definitions,
  3. interesting examples,
  4. one or two main results, and
  5. some nontrivial proofs.
Pictures and diagrams are welcome, as is historical information, but not to the exclusion of some serious math.

SCHEDULE:

  • Topic choices due by Oct. 11 (Thu);
  • Outline due by Oct 29 (Tue);
  • First draft due by Nov. 12 (Tue);
  • Practice talk to other students at least 4 days before presentation;
  • Practice talk to me at least 2 days before presentation;

TOPICS: There are many interesting and accessible topics in algebraic combinatorics. Here is an intiial list of possibilities. If you find one interesting, first see what you can find out on the web about it, including pointers to books and math papers. I'll also be happy to guide you to accessible places, but would also like to see what you're able to find on your own. (I expect to learn quite a bit!)

Note that some of these topics are much too broad to cover thoroughly and will need to be appropriately narrowed to a manageable project.

  • Abelian Sandpile Dynamics: There's a fascinating connection that's just been uncovered with Appolonian circle packings.
  • More on posets, which is a broad area of combinatorics. One topic that has a similar flavor to the work we've done on walks and posets is the subject of "Differential Posets".
  • More on Markov chains or discrete dynamical systems;
  • Basics of finite group representations;
  • Integer partitions and their identities;
  • Dilworth's theorem for posets and applications;
  • De Bruijn sequences and universal cycles of permutations;
  • f-vectors of polytopes and the Kruskal-Katona theorem;
  • Mathematics of juggling;
  • Catalan numbers;
  • Euler numbers & Eulerian polynomials;
  • Generating functions;
  • Permutation statistics;
  • Dyck paths;
  • Lattice-path method (of Lindström & Gessel-Viennot) & determinants;
  • Umbral calculus;
  • Using linear algebra to prove Fibonacci identities;
  • Perron-Frobenius Theorem and applications;
  • Something else (check with me before going too deep)

SOURCES: Beyond standard googling, there are a number of useful sources that you may want to check explicitly for information about your topic. Many are available on the web, but may be protected by pay wall unless you visit them from a location on-campus (or using VPN) or through the library website.

  • The arXiv of Math and Physics preprints has become a major repository of papers, both making them available before formal journal publication and maintaing their availability in some form even if one does not have access to all possible journals.
  • The Mathematical Association of America Monthly publishes articles that are often more accessible than those in the research literature. Back issues are availabe through JSTOR.

PRESENTATION: The presentation should last 20 minutes (which is both shorter and longer than it first seems). Rather than reading your paper, you need to think about how to present the material most effectively in an oral presentation. Technical details and detailed proofs should be avoided in favor of giving listeners a feel for the subject and why they should find it interesting. This is not a license for sloppy statements, but if you need to be vague about something too technical to present in a short amount of time, be clear that this is what you are doing. .

You should probably use some sort of projector, either overhead transparencies or from a laptop, although writing on the blackboard might be appropriate instead. Examples and visuals are great, as are props (if appropriate). Handouts can be helpful to many, and can give you a place to put technical details or long-winded statements that you don't want to take the time to write down.

Everyone must practice their presentation in front of other students (not necessarily from Math 3094) and get feedback from them. The presenter should keep track of this feedback and then forward it to me. Afterwards, the presentor will practice it again for me in my office and get feedback from me. This will make your final presentation to the entire class much more polished than if it were your first time.

Since giving and receiving feedback is a skill that benefits from explicit teaching, here is a handout on feedback that everyone should read before giving (or receiving) a practice presentation.


Back to the Math 3094 home page.

 
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