Project

There is a project which will require

A paper to be finished November 12
A group presentation of the papers in the last week of class

Paper assignment

Each of you has been given a chapter based on the topic you chose. You will be writing a paper that describes the main point of the chapter to another member of the class in a way that is more easily understood than the chapter, and in your own words. The paper should be typed.

Before you begin writing, you should determine what the main result is and work out what you need (definitions, examples, etc.) to state, prove, and understand the main result. You should also think about what consequences might be used to help motivate why one should care about the result.

The paper itself should have an introduction which motivates the topic and gives an idea for what the main result says. If you can actually state the main result without needing definitions, even better. Then you should have a section that deals with the basic definitions and results you will need to prove your result. You do not need to include proofs of results we have proved in class, but if you need a result to prove your main theorem, state it here. Examples are of course appreciated to help understand definitions. Next, you should state and prove the main theorem. Be sure to write in complete sentences, even if you display key ideas and equations. The last section is somewhat more open-ended. You may wish to include some consequences or corollaries of your result, or what one might want to consider next, given the main theorem. Alternatively, you may wish to give a sufficiently complicated example that illustrates some of the main ideas of the main theorem.

Writing software

While one can use most word-processing programs for typing in mathematics, those of you interested in pursuing careers in the sciences (or related fields) may want to consider learning some version of LaTeX. This program has the advantage that documents written look right, even if they are not necessarily mathematically correct.

If there is general interest, we may spend a class going over the basics of this editing language, including information on how to obtain and install the (free) software.

Here are the notes we used in class during the Latex presentation: .tex, and .pdf.

Possible Topics:

Turan's Theorem and extremal graphs

Turan's theorem says that if a graph on a fixed number of vertices has sufficiently many edges, then it contains a complete graph. See where this kind of a result leads.

Dilworth's Theorem and extremal sets

Dilworth's Theorem is a result about posets (partially ordered sets). It has some nice consequences concerning subsets of a set.

Flows in networks

A flow network is a directed graph with edge weights that tell you how much can flow through and edge. What kinds of things can one say about these systems?

DeBrujin Sequences

DeBrujin sequences study how to place a list of things in order in such a way that at each step you change just a little from the previous step.

Permanents

Permanents are like determinants, but different. Here are some combinatorial things to say about them (requires linear algebra).

Quadratic forms and graphs

Quadratic forms are a useful tool throughout mathematics. See what one can say about them using graph theory (requires linear algebra).