Section Group Project
A quarter of your Mathematical Engagement score in this course is the completion of a group project, in which you will explore an application of abstract algebra to your field of study. Here are the details about that assignment.
- Group size
- You can work in a group of size 1 to 4 (3 is recommended).
- Topic
- Pick a topic that is of interest to you from those suggested below, or one you propose with instructor approval.
- End product
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The result of the project will be a short paper (approximately 5 pages) in which you capture what you have learned about your chosen topic and how it connects to the abstract algebra content we have studied this semester.Note that the syllabus said that the project would also include a class presentation, but this will not be included in the expectations for the project.
- Due dates
- The final paper is due no later than the final exam for our class, namely Monday, December 8 at 1:30pm. There will be additional checkpoints due prior to then to help keep you on track. These will be posted on Canvas.
- Allowed resources
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You are encouraged to use resources you find online or through the library. All sources must be cited in the paper. Generative AI can be used for initial investigation of the topic, but you should use this skeptically as it is often wrong about mathematics (and you will still need to cite actual sources for what you find).Under no circumstances are you to use AI to write your final paper (beyond built in spelling and grammar checkers of your text editor).
Suggested Topics.
If you have heard about an application of abstract algebra besides those listed here, please check with your instructor to ensure it is appropriate.
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Teaching secondary algebra: The goal of this topic is to think deeply about the connection between abstract algebra and how teaching algebra at the secondary level are related. You should find some papers about this topic and also include your own insights about how this class might influence the choices you make as a teacher.
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Coding Theory: When you transmit a message, it is reasonable to expect the introduction of some errors. Is there a way to “pad” your message so that even if some of it can be lost (or randomly altered), you can still reconstruct the original message? What is more efficient than just sending the message 10 times? And how do polynomials help us in this process?
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Knot Theory and Braids: Knot theory studies the different ways a rope or ropes could be tangled up. How can you tell different knots apart? We would consider the much simpler question of understanding braids (which can be transformed into knots) using group theory. This is one example of a larger topic: group representations.
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Lattices and Boolean Algebra: Rings are sets with two operations. When studying sets, you might also consider the operations of union and intersection. Are these like plus and times for rings? Or are there other axioms that govern how this operations work? Boolean algebras are a generalization of the set (or logic) operations, and provide the basis for the algebra of electrical circuits. Besides the application, this is a nice example of other algebraic structures beyond groups and rings.
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Wallpaper Groups: How many different wallpaper patterns are there? What can abstract algebra tell us about translations, rotations, mirrors, and other symmetries? Oh, and crystals!
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15 Puzzle: This physical puzzle consists of 15 squares tiles that can slide around a 4 by 4 grid (so one open space that an adjacent tile can slide into). Here’s the question: if the tiles fell out and you randomly put the grid, what is the probability that you would be able to slide them back into their correct position? How can groups help us some the puzzles that have a solution?
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Rubik’s Cubes: How can groups help us solve a Rubik’s cube? This will serve as an excuse to investigate some interesting ideas in group theory. Plus: learn to solve a Rubik’s cube!
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Combinatorics - Counting bracelets: How many different bracelets can you make using 10 beads that come in 4 different colors? Careful: two bracelets are the same if it is possible to flip or rotate them to line up the colors. How many different ways can you paint to faces of a cube using 3 colors? We will see that groups of symmetries help us answer these questions. Along the way we will explore group actions and orbits and stabilizers.
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Cayley Graphs: Here is another interesting connection between algebra and combinatorics, this time dealing with graph theory. For any group, we can define a graph that describes how that group “acts” on itself. Given any graph, we can ask what the group of symmetries (really automorphisms) are. There is lots to explore here, including lots of pretty pictures.
