Math 31: Topics in Algebra

Summer 2013

Projects

As part of this course, I would like you to have the chance to explore some mathematics on your own. With this in mind, you will be required to complete a group project. You should work in groups of 3 or 4 to put together a 15–20 minute presentation, which will be given at the end of the term. In addition, each member of the group will need to turn in a paper addressing their contributions and what they learned during the completion of the project. You are welcome to choose your own groups, and I can try to help in matching people up if necessary.

In your presentation, you should tell the class about an interesting mathematical topic related to abstract algebra. Some vague suggestions are:

• Learn about a more advanced topic in algebra which we have not covered in class, and give the class a synopsis of it.


• Tell us about an application of algebra that you find interesting.


• Research some of the history behind abstract algebra and the mathematicians who developed it.

I will have a list of specific topics that I can suggest to you, or you can come up with a topic on your own and ask me if it is appropriate. I can also direct you toward sources that you can use to research your topic. In addition, Shirley Zhao (from the Kresge Physical Sciences Library) has offered to give you some research tips during a couple of the x-hours.

To make sure that everyone is on top of things and that the topics are acceptable and relevant to the course, I will set some deadlines for the project:

• July 9: Give me your choice of partners, or tell me that you would like to be matched with someone.


• August 2: Choose the topic for your presentation. Give me a written proposal (as a group) containing a tentative outline of what you plan to talk about.


• August 9: Turn in a written outline (as a group) with a summary of your presentation and a description of the main points that you intend to cover. Also include a list of references.


• August 13–16: Give your presentation in front of the class.


• August 19: Turn in your individual papers.

Below are details regarding the different components of the project.

• First (rough) outline: This should, first and foremost, tell me what you topic is and how you plan to approach it. You should give a brief description of what you plan to address in your presentation. It's fine if the details are somewhat tentative at this point. This need not be long—the purpose is to tell me what you plan to research so that I can make sure it is appropriate.


• Second (detailed) outline: At this point you should have a pretty good idea of what you plan to accomplish in your presentation. This outline should give a detailed summary of your topic, including the major points that you intend to address. For example:
  
  
  • If you are investigating an advanced topic in abstract algebra, state some of the definitions and theorems that you plan to discuss.
  
  
  • If you are studying an application of algebra, describe the aspects of algebra that are used and how the topic relates to the class.
  
  You should also include a list of references that you have been using in your research. This outline should be more detailed than your first one, but it does not need to be overly formal (a bulleted list is fine). Its main purpose is to give me a good idea of what you plan to address in your talk, and to inform me of the references that you've been consulting.


• Presentation: This is your chance to tell the class (and me) about your topic. Your talk should be roughly 15–20 minutes long, and all the group members should contribute in some way. You are free to give the presentation via whatever medium you find most comfortable (chalkboard or slides). There may also be time for you to field questions from me and your classmates at the end.
  
  Below is the tentative schedule for the presentations:
  • Wednesday, August 14:
    Cryptography (Mimi Rogers and Jennifer Jin)
    Group Theory and Music (Laura Bergsten, Robbie Bhattacharjee, and Patty Neckowicz)
    Rubik's Cube (John Cofer, John Conley, and Asher Mayerson)
  • Friday, August 16:
    Boolean Algebra (Emmanuel Blankson, Lola Ojabowale, and Jinjae Park)
    The Riemann Hypothesis (David Bessel, Carly Christian, Jane Karpis, and Shelby Schier)
    Carl Friedrich Gauss: The Prince of Mathematics/Titan of Science (Chester Brown, Luke Rossi, Sahil Seekond, and Hannah Wang)


• Individual report: Each group member needs to turn in a written report to accompany their project. In it, you should discuss some of the mathematics that you specifically contributed to the project. (For example, if your group chooses to divide the responsibilities, you could discuss the specific topic(s) that you were tasked with researching.) It is possible that there may be some overlap between the topics addressed by the different group members, but your paper should ultimately be written by you on your contributions. You may also want to address some (but not necessarily all) of the following questions:
  
  
  
  • How did you become interested in your group's topic?
  
  
  • What did you feel that you learned or gained from the project?
  
  
  • What aspects of the topic did you find particularly interesting?
  
  
  • What are some specific connections between your research topic and the ideas that we have discussed in class?
  
  
  The paper should be approximately two (2) pages, and it must be typed. It is due by the second-to-last day of class (August 19).

The following is a breakdown of how the overall project grade will be computed. We will discuss specific rubrics for the presentation and paper as the term progresses.

When assigning a grade to the presentation, I'll be looking for the following things:

• Did you draw clear connections between your topic and the material that we've covered in the course?
• Was everything explained clearly and thoroughly?
• Did all the group members contribute equally (more or less)?
• Did you demonstrate a good understanding of the material that you've chosen to cover and the algebra that was involved? (This may include the presentation itself and any questions that I might ask you afterward.)

I will base the grade for the paper on the following things:

• Was the paper written clearly, using proper English (including grammar, punctuation, spelling, etc.)?
• Did you demonstrate an understanding of your group's research topic, and specifically the parts that you researched?
• Did you clearly convey your contributions to me?