Math 109: Writing-In-Major assignment information

WIM assignment info: Draft due February 29, final version due March 9.

The WIM Assignment is to write an exposition of a topic related to group theory. The target audience is someone at a similar stage in a similar class. The length of the exposition should be around five pages.
Although an "official" draft is due February 29, I encourage you to seek feedback -- either from myself, the CA, or from your classmates -- earlier than that.

Topics

Recommended topic: symmetry groups of the Platonic solids, with emphasis on the case of the icosahedron. Your goal is to describe, with proof, the symmetry groups of the five Platonic solids. This is discussed in Chapter 8 of Armstrong, and you should flesh out the discussion there.

Other possible topics:

  1. Point groups and symmetry of molecules

  2. Elliptic curves and their use in cryptography:
    A good core topic here is to explain elliptic curves over the real numbers : what the group law is, with examples, and a proof of associativity. You might add a brief explanation of why they are called "elliptic," or why elliptic curves over the integers modulo a prime are of use in cryptography. A good reference is the book of Silverman and Tate, "Rational points on elliptic curves," which is on reserve at the math library. You can find many other references by Googling, some elementary and some less so.

  3. The group theory of the Rubik's cube.
Although these other topics (may) sound like fun, you should do this only if you know what you are doing and are confident about the material. (2) and (3) in particular require some prerequisite knowledge that we have not covered in the class, so you have to be willing to put in more work. If you are interested in them, see me to discuss references.

How to write it

When writing a work of mathematical exposition, you are striving for a mix of precision and informality. That is to say: you need to informally communicate the main ideas, while, at the same time, giving precise proofs.

More practically:

  1. Introduction: Start with a general description of the topic, and in particular the mathematical problems that you will discuss. You should say why you think it is interesting, and what the main ideas involved are. It's often good to state the central theorem(s) that you will prove precisely.

    You should do this with a minimum of technical jargon. Choose very carefully what to include and what not to include: if the reader stopped reading at this point, would they have understood the essential points?

  2. In the body of the text, solve the problems that you've raised in the introduction. For example, if you stated a Theorem in the introduction, prove it here.

    This part should be as self-contained as possible, so as to be accessible to (e.g.) someone at a similar stage in a similar class, but not necessarily using exactly the same text. Thus "by a theorem from class" is not a useful way of describing a result; state the result precisely, and give a reference.

  3. Conclude by summarizing some important points and (if possible) explaining some interesting problems that are related to your paper.
Finally: Proof-read! (Better, have somebody else proof-read.) I find it helpful to read my writing out loud to myself. Check for undefined symbols and notation as well as mathematical correctnesss. Also check for clear English: write in complete sentences, and do not use symbols for "implies, there exists, for all," etc.: write them out.

I'm very happy to discuss anything related to this. I am also happy to look at your drafts and give some comments. Don't hesitate to email me with questions or drafts; I will send back some quick comments.

Akshay Venkatesh
Department of Mathematics Rm. 383-E
Stanford University
Stanford, CA
email: akshay at stanford math