Math 281 Modern Algebra II

Your chosen topic is not expected to be original research. The paper and talk should be expository in nature, meaning you will simply need to choose a topic related to abstract algebra. Ideally, your topic will be something you are personally interested in, and which we have not already seen in class.

Your talk could provide an overview of a particular application of a topic from abstract algebra. There are a variety of immediate connections between abstract algebra and computer science, cryptography, physics, chemistry, and even the arts. Several of these applications are highlighted in the textbook. Or, you could choose to discuss an important theorem or topic in algebra that we have not covered in class, along with a brief look at its mathematical significance and historical context.

If you can answer "yes" to each of these questions, you have probably found a good topic:

• Is there a clear connection to abstract algebra and your topic?
• Can you explain why your topic is of mathematical or historical significance, or why would be of interest to the audience?
• Can you find several references to learn more about your topic?
• Will it be possible to explain your topic, including any necessary background material, in a 15 minute presentation?

In some rare cases, a topic that does not fit under the abstract algebra category may be approved. The topic should still be an advanced concept in mathematics, such as the Brouwer fixed-point theorem.

Your abstract should be considered an advertisement for your talk. Most abstracts are 5-6 sentences in length, but this is not a hard rule. There should be enough detail to give the audience an idea of what your talk is about, but not so much detail that your topic is obscured. You also want to avoid giving away your entire talk - reserve the punchline for the actual event.

Include the title for your talk with your abstract. References should not be included, unless the topic is about the reference itself, and even then you'll want to leave out publication dates and the like.

An abstract falls under the usual rules for mathematical writing. Your abstract should be formally written, using "we" instead of "I".

Brief definitions should be included for words that are likely to be unfamiliar to your audience. For this talk, you do not need to include definitions of any terms from class. However, if you plan to give a talk on nilpotent groups, you may want to quickly explain what they are.

Use mathematical notation and symbols sparingly, if at all. Use words to describe what you will talk about. There are two reasons for avoided symbols: your abstract will be easier to read, and you will not need to worry about how your abstract will appear in print. LaTeX is fine for class, but should be avoided when submitting an abstract to a conference.

Above all, your abstract should make the reader want to know more and come to your talk!

You will have a total of 20 minutes for your talk (including audience questions), so aim for a topic you can adequately cover in approximately 15 minutes.

Your talk should not be a lecture. Your audience will not be tested on the material, so there is no need to provide every detail. However, you should provide enough depth so that your audience stays connected and listening to the talk. Talks in mathematics generally have little audience participation until the question period at the end, though you may choose to ignore this convention for our class.

You will be free and encouraged to use the chalkboard, projector, handouts, or any other materials that could assist your talk. No particular media is required, but slides can be helpful for long definitions, formulas, and images.

Along with your talk, you will be submitting a conference paper for publication in The Proceedings of Math 281. A draft of your paper will be due April 11, with the final version due Friday, April 27. The draft will be returned with corrections and suggestions, but will not be used for grading.

There is no required length for your paper, but it should likely be at least one page and no more than five (single spacing, 1-inch margins, reasonable font size). The paper will usually include more detail than your talk, and should be written with the usual formal tone. Again, use symbols sparingly. It is better to write "Let \(x\) be an element of \(G\)" rather than "Let \(x \in G\)".

Your paper will generally be written at a higher level than your talk as well. Talks are usually more accessible, and are meant to be enjoyed by audience members that may not be experts in the field. The reader of a paper is likely much more interested and knowledgable in the topic, and the paper may be written accordingly. As a rule of thumb, the talk is for your classmates, and the paper is for your professor.

The final version of the paper must be typed and include all references. The paper should be submitted as a pdf. You are encouraged, but not required, to use LaTeX.

Any references used for your talk or paper should be included. There is no standard format for references in mathematics, as the formatting is usually at the publisher's request. Peer reviewed publications and textbooks are strongly preferred over websites such as Wikipedia (peer reviewed online journals are an exception). These sites can be a great starting place, but follow the citations provided on the page to learn more and make sure the information is correct.

References should be listed at the end of your paper, in alphabetical order by the first author's last name. References are usually numbered with brackets.

Citations in mathematics are used for efficiency. If your paper makes use of a lengthy proof or other result you found in an external source, you do not need to repeat the entire passage. Simply mention in your paper where the proof or result could be found.

Example:
Let \(n\) be a finite group with order \(n\), and let \(x\) be an element of \(G\). By Lagrange's theorem [2, p.147], the order of \(x\) is a divisor of \(n\).

References:
[1] Garrett Birkhoff and Saunders Mac Lane. A Survey of Modern Algebra (5th ed). Macmillan, 1996.

[2] Joseph Gallian, Contemporary Abstract Algebra (8th ed). Houghton Mifflin, 2012.

• Mac users: download TeX from the MacTeX
• Windows users: try MiKTeX, but other distributions are available.
• Overleaf has several templates for reports, slides, and more to get you started.
• ShareLaTeX has some quick tutorials to help with the basics.

• College Mathematics Journal (MAA): You won't be able to access the articles unless you're a member of the MAA (there's a student discount!) but you can read through the abstracts. If there's an article you're interested in, just let me know and I'll get it for you.
• American Mathematics Monthly (MAA): MAA again, so if you find an interesting article, I'd be happy to get it for you.
• Mathematics Magazine (MAA): Just join the MAA already.
• Harvard Mathematics Review: There's only six issues, but each is filled with articles written by and for undergraduates.
• Rose Hulman Undergraduate Math Journal: Lots of issues to look through, though many of these will be in applied math areas (that's still a possibility for a conference talk, you'll just need to ensure it's related to algebra).
• Involve: A math journal that publishes papers with at least one undergraduate author. These may be a little higher level, but a good summary of an article would be a great talk.
• arXiv.org: Ok, you probably won't find your topic here, but have a look at some great (free) math!