Term paper, first draft

Due: Nov 20th, 2020

Write a first draft of your term paper.

Purpose

The purpose here is two-fold:

  1. Have an opportunity for feedback and suggestions.

  2. Making sure you stay on schedule to finish a term paper without a big rush at the end of the semester.

Requirements

Given the realities of an academic semester, specifically this academic semester, your term paper first draft is allowed to be an incomplete draft. Here is what is required:

  1. Your paper should have a title and identifiable topic/theme.

  2. Every section of the paper should be present (should have at least something in it), and at least one section should be nearly complete.

    • Recommendation: The (nearly) complete section should be one of the main content sections, with the main theorems or highlights of your paper. It should not be the Introduction or a the background section. Those are important too, but not a good place to start your writing, because they can change a lot if the main sections change. Therefore I recommend to get the main sections figured out and written first.

  3. There has to be at least one theorem statement and at least one proof.

    • This could be a smaller result such as a lemma or corollary but it would be best if this were one of the main results.

    • At this stage the proof can be incomplete. But if possible, for my sake and also for your own sake, give some kind of indication of where are steps that you plan to add more but just haven’t gotten around to it yet, as opposed to steps that you plan to omit from the final writeup (e.g., with a citation).

  4. There has to be a bibliography with at least one reference, with proper formatting. (See “How to Bibliography” guide posted in BlackBoard.)

  5. Use LaTeX to produce a PDF with reasonable formatting, similar to homework.

    • Readability: Please use reasonable margins (at least 1 inch), a 12-point option such as \documentclass[12pt]]{amsart}, and \linespread{2.4} to produce something like double-spaced lines.

    • LaTeX Environments: Please use LaTeX environments such as \begin{theorem}...\end{theorem}, proof, etc. See for example Overleaf’s tutorial on theorems and proofs.

Advice

Finally, some advice, which you are free to ignore. A normal part of the writing process is to realize that your plans and goals were too ambitious. If you start to feel that you have taken on too large of a writing task, some options that you might consider include:

  • Focus on a special case of your main result. For example, instead of \(n\) dimensions, just deal with the \(1\) or \(2\) dimensional case. (Better yet, start with the “trivial”/”easy” cases, and then focus on the first non-trivial case.) You might just briefly discuss the more general case, or just state it, or even just give a citation to it; or maybe not get into it at all.

  • Omit some technical or difficult things. Don’t skip over the main points of proofs, or the important things that are the reason your topic is interesting. Keep those important steps. But there might be other steps that can be cut while still conveying all the important points.

  • Pick a running example. This can be used to illustrate or demonstrate some results, or some steps, when the general case is too difficult. This might require two or more examples.

  • You can expect something from your reader. Don’t feel that you have to explain every single thing to the reader. You can expect them to know something. To play it safe, you can “recall” specific facts that you need.

  • Don’t get bogged down in introductory or trivial things. It can make a very good paper to include some introduction, context, history, etc., and also to explain a simple or trivial case, in order to illustrate by contrast what makes other cases difficult or non-trivial.

    For example, in \(n=1\) dimension, all ideals are principal, all polynomials factor into linear factors (if the field is algebraically closed), there’s only one monomial order, etc. “Proving” your main theorem in this case can help the reader appreciate the non-trivial cases. However, don’t get bogged down in this. If the main point of your paper is the non-trivial case, then keep that in mind, treat the easier case as just setup for the main point, and get to the main point.

    The same goes for the introduction. Take as long as you need to set up your paper and to establish an interesting topic. But don’t take any longer than that.