## Math 332 Projects, Spring 2015

For your final project in Math 332, you will write a 5-10 page paper on a special topic in algebra. Using the techniques we have developed throughout the term, you will explore a topic slightly outside the standard curriculum. Your task is to choose a topic, understand definitions, basic theorems, and proofs surrounding that topic, and then write a document that explains that material to your fellow students. You don't have to prove everything you write about, but the document should contain precise statements and some proofs.

You are encouraged to consult the instructor in choosing your topic. Below you will find a nonexclusive list of potential topics you could consider studying. But first, deadlines:

• Topic approval: Friday, April 3. Submit a one paragraph description of your topic via email, briefly indicating what you hope to cover. You will receive feedback on whether the project is appropriate.
• Draft: Wednesday, April 22. Submit a draft, at least 2 pages long, in class. You will receive comments on the draft that will help in the preparation of your final paper, so writing more than 2 pages is encouraged. Draft submission is absolutely required.
• Final paper: Friday, May 1, at the start of class.

Potential topics are many and far-ranging. The only requirements are that (a) the topic interests you, (b) algebra (groups, rings, etc.) plays a significant role in the definitions or theorems related to the topic, and (c) you can reasonably exposit the material to your peers in 5-10 pages. Below are a few possibilities. Use the internet (google and MathSciNet) and the library to try and track down information, and talk to me if you are having trouble finding good sources.

• The Burnside ring. Given a finite group G, the Burnside ring of G encodes data about finite G-sets. It's related in important ways to equivariant homotopy theory and quadratic forms.
• Groups acting on trees. If a group G acts on a tree X, then G is isomorphic to the graph fundamental group of X/G. The theory is due to Bass and Serre, and Serre's 1977 monograph Trees is the canonical source.
• The abelian group structure on an elliptic curve. Elliptic curves are the roots of special types of cubic polynomials in two variables. Somewhat miraculously, the points of an elliptic curve admit a geometrically defined (tangent lines, reflection, etc.) group structure.
• Monads. Many of the notions of algebra are in fact examples of "algebras over monads." Monads are a categorical construction consisting of an endofunctor and two natural transformations: a "unit" and a "multiplication." Monads are also essential tools in modern computer science (especially functional programming and big data, cf. algebird).
• Basic algebraic geometry and the Nullstellensatz. Algebraic geometry studies roots of polynomials from a geometric perspective. To each ideal J in a polynomial ring over a field, it associates a variety V(J) consisting of common roots of elements of I in an algebraic closure of the base field. One can also form an ideal I(V) from any variety V. Hilbert's Nullstellensatz determines the value of I(V(J)) and is a fundamental tool in algebraic geometry.
• Normed division algebras over the real numbers. Divison algebras (possibly non-associative) over the reals which also have a multiplicative "norm" are very rare. In fact, there are only four of them: the reals, the complex numbers, the quaternions, and the octonoions.
• Braid groups. You can form a group out of n-strand braids. It's kind of like the symmetric group on n letters, but it's infinite! This is an interesting class of groups with many pictures and obvious relations to knot theory.
• Group theory and puzzles. There are a lot of mediocre things written about groups and puzzles like the Rubik's cube. Check out Simple groups at play by Igor Kriz and Paul Siegel for something more interesting.
• Sporadic simple groups. Speaking of which, the theory of sporadic finite simple groups is rich and mysterious. The monster group? The baby monster? The Mathieu 12 group? These are interesting and deep objects.
• Representation theory of symmetric groups. Representation theory studies groups via matrices. Irreducible complex representations of the symmetric group are in one-to-one correspondence with a class of simple diagrams called Young tableaux. of
• The Temperley-Lieb algebra. The Temperley-Lieb algebra is a fascinating ring whose relations are based on planar string diagrams. It has fascinating connections with the representation theory of braid groups and "percolation" in statistical mechanics. Check out Knots and physics by Kauffman for a nice presentation.

It is recommended (but not required) that you write your paper using the LaTeX document preparation system. You are welcome to use the following template in preparing your writeup.

Please do not deviate substantially from the standard formatting imposed by the above template (letter paper, 11 point font, 1.5 inch margins).

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