Term paper, topic proposal
Term paper, topic proposal
In this assignment you will choose a topic and one or more sources for your term paper.
As a reminder, from the syllabus, this paper is:
a term paper (6–15 pages) about a topic of the student’s choice within algebra or algebraic geometry.
Assignment
In this assignment please look at some books and papers in order to identify a topic for your term paper.
You will write 0.5-2 pages to describe the main topic of your term paper, and address at least some of the following questions:
- What will your paper be about?
- Why is it an interesting topic? Why are you interested in it? (How will you make it interesting for the reader?)
- What will be the “main takeaway” or main point of your paper? What parts and technical details will you skip over, to get to the main point?
- What one or more high-quality sources will you use for your paper?
Purpose
The purpose of this assignment is to help you write your term paper, in these ways:
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You will take your first planning steps for your term paper. By starting your planning now, instead of at the last second, you’ll have plenty of time to work on your term paper with a lot less stress.
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I’ll give you feedback on your topic proposal.
Formatting
- Please write your responses in a PDF file created using LaTeX.
- Upload the PDF to Canvas. (Not Gradescope. I am trying Canvas as an experiment.)
Grading
You’ll get full credit for writing and submitting a term paper topic proposal. The purpose of this assignment is to help you start your term paper writing process, not to grade your work.
Topics
The chosen topic must be at a similar level to the course and related to abstract algebra and/or algebraic geometry. Topics that are directly related to course material are preferred, but not required.
You may not write a paper about your own research unless you discuss it with me first.
You are invited to freely select a topic of your own choice. Choose something you are interested in, something that excites you! If you don’t know where to start, some suggestions include:
- Bezout’s theorem and Cayley-Bacharach theorem
- Faugere’s algorithms for Gröbner bases
- Using algebra to solve graph coloring problems, sudoku, and other puzzles (e.g., chessboard puzzles)
- Eigenvalues of tensors
- The game SET, the cap set conjecture, and the polynomial method
- Symmetric polynomials
- Fibonacci polynomials
- Algebraic games: A game on Noetherian rings, A game based on the Euclidean algorithm (can you design a polynomial version?)
If you’re able to attend some TATERS talks, you might learn about some very exciting topics from our visiting speakers.
You may also choose to write a term paper focused on biography or history, see below.
In addition, there are also some excellent topics in the course textbook and other recommended sources, see below.
Biography or history
You may choose to write a term paper that emphasizes biography or history. In this case you have to focus on the mathematical parts of the biography/history, and specifically mathematical parts that are related to course content. For example, you could choose a mathematician, and give a short general biography, an overview of their mathematical accomplishments, and a focused presentation of one or two specific mathematical accomplishments related to this course.
Some questions to consider are:
- How did these mathematical ideas arise in the context of their time?
- How did this person come up with these ideas?
- What effects or influence did the ideas have? Decide if you want to present the mathematical ideas in the original sources, or a modern take on the ideas.
The MacTutor History of Mathematics Archive http://mathshistory.st-andrews.ac.uk is a very good source for mathematical biographies, and it also has references to further sources.
Please do not write a biography of any living person. Suggested subjects for biographies in this course include:
- Emmy Noether
- J.J. Sylvester
Sources
Recommended journals include the American Mathematical Monthly, College Mathematics Journal, Mathematics Magazine, Math Horizons, Involve: a journal of mathematics, and What’s Happening in the Mathematical Sciences. Also, many excellent articles are available from the MAA Writing Awards web page, https://www.maa.org/programs-and-communities/member-communities/maa-awards/writing-awards.
From the course textbook, you can find topics from textbook sections not covered in class. There are also projects in Appendix D of the textbook.
In recent years an amazing number of books have appeared with lots of topics relating computational algebra and algebraic geometry to statistics, machine learning, areas of science, etc. You might be able to find some interesting topics in these:
- Pretty much any single chapter in the Mateusz-Sturmfels book
- Algebraic statistics from the books of Drton-Sturmfels-Sullivant (pdf) or Sullivant. (These might require some study, and some parts of them might be a little bit more advanced, but if you’re interested in statistics, data science, information geometry, or machine learning, then these might be interesting topics for you.)
- Applications of Polynomial Systems
Allowed sources: You should only use sources listed in MathSciNet (https://mathscinet.ams.org), or listed above. (For example, the student-oriented journals listed above–College Mathematics Journal, Mathematics Magazine, etc.–aren’t listed in MathSciNet, but they are excellent, reliable sources of high-quality information. However, “research journals” that aren’t listed in MathSciNet are not necessarily reliable sources of high-quality information.)
If you have questions about other sources, you can ask me.
Web pages and videos can be excellent aides to help learn and understand a new topic, but they may not be appropriate sources for citations in written work. One exception is the MacTutor History of Mathematics Archive http://mathshistory.st-andrews.ac.uk, which is a good source and can be cited in writing. If you find other web pages or videos that you are interested in using for your term paper, you should try to look for a published article or book that includes the same information. (High-quality web pages and videos should include clear references to their source material.) Again, if you have questions about using a web page or video as a source, you can ask me.
Advice
Don’t pick something easy but boring (“the commutative rule for addition”). On the other hand, don’t pick something way too advanced (“motivic integration for perfectoid spaces”). Instead, pick something interesting and fun, something that you will enjoy learning about, and writing about–something that you want to share!
Importantly, you will have to choose a narrow slice of your topic to focus on. Don’t try to tell the whole story of a major subject; instead, choose one or two interesting examples and theorems. To write about an interesting but difficult theorem, skip over part of the proof, or focus on a simpler special case, or just roughly sketch the proof but instead give a detailed workthrough of an example. For instance, give the proof only for 1 or 2 variables, only for continuous functions, only for monomial ideals, or for some nice family of rings instead of for all rings. Find the right balance where you can tell a story that’s still interesting, but simple enough for beginning students to understand, within a 6-15 page scope.
The upshot is that it’s okay to choose a paper or book and only plan to cover part of it in your term paper! For example, if you are interested in algebraic statistics, then you don’t have to cover the entire book by Seth Sullivant, or even a whole chapter–you can choose a specific topic within that book to focus on.