# overview

You and your team will be asked to complete a project related to abstract algebra on a topic of your choosing. There are two main components of the project:

- Project write-up: your group will carefully write up your findings on a page of the course wiki,
- Project presentation: your group will make a 15 minute presentation to the class that summarizes your findings.

I want you to explore an advanced topic, summarize your findings in the write-up, and then teach the class about it in your presentation. Here is a list of questions that you should consider addressing:

- What is the topic all about?
- What is the history of the topic?
- How does it fit into big picture of mathematics?
- What is an application to science, another area of mathematics, industry, or art?
- What are some examples and non-examples that illustrate concepts related to your topic?
- What are some key theorems related to your topic?

# purpose

There are several motivations for making you complete a group project for this class, though the primary one is to give you a chance to explore some of the many interesting abstract algebra topics we will not have the chance to discuss in class. This is one of your major opportunities for creative engagement in the course, and you are encouraged to treat it as such.

Aside from the conspicuous mathematical content, however, this project is designed to sharpen your skills as an analytical thinker — to challenge your ability to both understand and transmit information. In addition, you will have the chance to hone your teamwork skills.

# topics

Each group has the freedom to choose their own topic related to abstract algebra, yet it should be something that your team finds interesting and that you can adequately report on in your write-up and during your project presentation. Here is a list of topics that I feel would be appropriate for your projects:

- Classification of the finite simple groups (already selected)
- Classification of the finite reflection groups
- Cryptography (already selected)
- Group actions and the Orbit-Stabilizer Theorem (already selected)
- Classification of the wallpaper groups (already selected)
- Galois theory
- Matrix groups & symmetry
- Solvable groups
- The Sylow Theorems (already selected)
- Application of group theory to science
- Lie algebras
- Representation theory (already selected)
- Develop a unit of lesson plans for introducing high school students to group theory

This list of topics is intentionally vague to allow you the freedom to fine tune your direction of study. You'll have to do some research on these topics in order to determine which one may be of interest to you. If you are having trouble choosing a topic, I will choose one for you. Each group must choose a distinct topic and the topics will be selected on a "first-come-first-serve" basis. Feel free to come see me if you want to know more about any of these topics.

Regardless of topic, both portions of your project should contain a discussion of an application to science, another area of mathematics, industry, or art.

# project components and timeline

To prevent you from cramming the entire project into the last 2 weeks of the semester, I have devised a timeline that each group needs to adhere to.

## group selection (due Friday, April 2)

Each group should consist of 3—5 students. Notify me via email who is in your group by **5PM on Friday, April 2**.

## topic selection (due Friday, April 9)

At least two members of your group must **meet with me** to discuss your project topic by **5PM on Friday, April 9**. Please make an appointment with me. Remember, each group must work on a distinct topic and the topics are "first-come-first-serve." When we meet, you should have two potential back-up topics, in case another group has already selected your first choice. I will keep you informed about what topics are being chosen.

## project outline (due Friday, April 16)

After you have chosen your group and topic, I will create a page on the wiki for your group to use to write up your project. By **5PM on Friday, April16**, I want your group to create a brief outline of your proposed write-up on your project wiki page. At a minimum, this outline should include:

- Title of your project
- List of potential references (you need a
**minimum of two references**) - Section headings (use "
`+ heading-name`" syntax for top-level sections) - Sub-section headings (use "
`++ sub-heading-name`", "`+++ sub-sub-heading-name`", etc syntax for sub-sections; if you look at the source for this page, you'll see what I mean) - Indicate where you will include examples to illustrate the concepts that you are introducing
- Indicate where you will discuss an application of your topic to science, another area of mathematics, industry, or art.

Despite being brief, this portion of your project will probably require the most groundwork. You will have to do substantial research to explore your topic and sift through the vast amount of information, so that you can decide what it is you want to include in your write-up. You will need to spend time searching the internet and combing through books in the mathematics section in the library.

## draft project write-up (due Friday, April 30)

Using your outline, create a draft of your project write-up. This is due by **5PM on Friday, April 30**. You should approach this draft as if it is your final version. I expect it to be close to perfect. My evaluation of this draft will constitute the majority of your grade for the entire project. This write-up should include:

- Complete sentences and proper grammar (Imagine you are writing a math book)
- Proper mathematical notation that has been typeset using $\LaTeX$
- Examples to illustrate the concepts that you are discussing
- Discussion of an application of your topic to science, another area of mathematics, industry, or art
- Properly formatted references (see this page for help with references; and of course, I will help, as well)

The write-up of your project should strike an appropriate balance of mathematical theory, examples, proof, and application. Of course, I understand that you will not be able to fill in all the details. Your write-up should be a summary of some aspect of the topic. If you need help including graphics, I will be happy to help.

## discussion of draft project write-up with instructor (due Friday, May 7)

During the week of May 3—7, at least two members of each group must meet with me to discuss the draft of your project write-up. This must be completed by **5PM on Friday, May 7**. Please make an appointment with me. During this meeting, I will indicate what additions, deletions, or improvements must be made (if any) to your project write-up. If you spent a sufficient amount of time on your draft, the amount of editing your group will need to do will be small.

## final project write-up (due Monday, May 10)

After meeting with me to discuss the draft of your project write-up, you should take my comments into account and make any necessary revisions by **5PM on Monday, May 10**.

## project presentation (Tuesday, May 11 & Thursday, May 13)

On **Tuesday, May 11 and Thursday, May 13**, groups will make a 15 minute presentation about their project. The presentations should be aimed at the students in the class and *not* at the instructor. You should be prepared to respond to questions that your classmates and/or instructor might have about your topic. You should also work to ensure that your classmates stay engaged during the class (you could ask them questions, have them solve simple problems, etc.). Imagine you are giving a 15 minute lecture on the topic.

**Important:** Groups should pay attention during each of the other group's presentations, as there will be questions on the final exam concerning the content discussed during the presentations.

The various due dates are also summarized on the schedule page.

# grading

**Beware of plagiarism!** Plagiarized write-ups will automatically be given an "F" on the entire project.

This project is worth 10% of your final grade. Scores for the projects will be determined for the group as a whole, though I reserve the right to give individual grades if team members are not actively and thoroughly engaged in all aspects of the project.

The total grade for the project is calculated as follows:

component | percent of overall grade |
---|---|

meeting of all deadlines | 10% |

project outline | 20% |

draft project write-up (including meeting) | 30% |

final project write-up | 15% |

project presentation | 25% |

# groups

Here are links to each group's project page on the wiki. Use these pages to report your findings in your project write-up.