About the Toolbox Essays
The toolbox essays are brief expositions written by students for students. You can think of these as brief, informal talks that just happen to be written down for the sake of current and future students. These are called "Toolbox Essays" since each of the essays is focused on "news you can use."
Although informal, there are some rules that we will follow to ensure some consistency of style. This format will also encourage proper scholarly presentation, e.g. proper use of title, abstract, content organization, and bibtex. This is all stuff that students need to learn before writing any real paper.
As a service to everyone, your paper will be posted on the web. An index to the papers will be posted on this page after the essays are complete.
Rules of the Game
- You should follow the form that I used in my little essay on the L^1 Maximal inequality (PDF, Latex, BibTexDataBase)
- Your essay should be from three to five pages long. If you find yourself wanting to write something longer, go ahead and write it --- but then cut it back to a maximum of five pages.
- The first goal of the paper is to create something that is useful. For this to be possible the paper has to be readable.
- Hints for readability:
- The mathematics must be correct. Nothing makes a mathematical paper more illegible more quickly than an error in a definition or a formula. Often one cannot read past the first error with any honest understanding.
- Choose good notation.
- Make equation displays attractive.
- Be very precise in the use of your language.
- Pictures and diagrams are GREAT!
- Polish your prose. After you have said what you want to say, go back over your essay and make sure that a reader who has not been where you have been can still understand everything perfectly.
- If you want to give "intuition" do so after you have given your basic message in a clear and complete way. Simply being clear often makes the struggle to provide intuition irrelevant.
- Another version of last rule: Forget motivation! Almost no one does it well, and it is especially perilous for beginners to try.
- Avoid pretentiousness.
- Avoid empty assertions. Don't say something is useful; show how it can be used. One of the greatest rules for writers is "Show, don't tell".
- Expository Finesse? Remove all adverbs. Remove all uses of the passive voice. Say "we" and don't worry about it. Learn the rules of punctuation. Do not use a mathematical symbol such as "=" or "<" as the only verb in your sentence. Finally, don't fail to rely on the "If ... then" structure; it is your friend, and it will get you out of many tough spots.
- Always, always, always --- did I say always? Keep your reader in mind. Papers are not written for file folders, they are written for people. Show your essay to a friend and let him read it alone without coaching. Ask him where he got stuck. Go back to that spot and improve your explanation. Repeat as many times as your friend will tolerate.
- Hints for "Quality"
(Based on Past Essays).
- The mathematics must be correct! This seems so obvious, but a very substantial fraction of the early essays have had major mathematical blunders --- sometimes of a truly conceptual kind, not just a computational or technical kind. Having a team should make it easier to avoid errors, but sometimes in a team there is too much "division of labor" and sometimes there can be a folie a deux where two people support each other in a delusional belief.
- Each team member needs to read carefully and skeptically. Errors can't be permitted to remain; they must be rooted out as soon as they are discovered, or the project turns to mush.
- The essay should make perfectly clear what innovations have been made and what is being carried along from other sources. Again, this seems obvious, but even a majority of the phase one essays have made this mistake. If you say you have "simplified" a proof, then say "how" and "where."
Hints on Finding A Topic
Believe me, I know that it is not easy to find a suitable topic. You should think about several possibilities before setting down to write. You should first plan your work with paper and pencil. Draw pictures. Think of order of presentation. Plan your essay like a talk; visualize the person that you are addressing.
The best way to come up with a topic is to do some reading, have a personal insight, and have the desire to explain that insight to someone else. In that case, the paper almost writes itself.
I will suggest some topics, but you can guess that all along I have been dishing out bits and pieces of such essays. I'll continue to add to the list --- both here and in class.
- "Variations on the Flop" (one was on the 530 final that many did not see)
- "Variations on the Characterization plus Compactness" argument (ditto)
- A cool characteristic function: X is integrable but ...(ditto --- more or less)
- Bessel's inequality: Probability arguments without independence
- Riesz products and Pathological CHFs
- Vitali-Hahn-Saks --- "New Measures from Old"
- "What Pros Known About Uniform Integrability that You Don't." WikiLink
- "Bounds on Log(1+x) and Why I Care"
- "Extreme Points and Hoeffding's Inequality"
- "Why Lazy Random Walk is Nicer than Simple Random Walk"
- "Feller's Proof of the Discrete Arcsine Law --- The Essential Steps"
- Erdos-Feller-Pollard: An Example of Characterization plus Compactness.
- The Master Theorem and CS Recursions: Refinements and/or Applications
- 'Polynomial Recursions and Carleman Linearization: One Trick from Several''
One of the nicest ways to get a useful essay is simply to ask yourself a question about our work in class, a proof in the text, or an argument in an original paper. Stay with the question until you dig up something that is worth explaining to someone else. This is actually pretty easy, once you get the hang of it.
At the end of the day, you have to pick something and live with it. Rain or shine.
How Hard Is This?
It is harder than you think to write a three to five page paper that is worth the time of another person to read. What makes this possible is that the paper can offer distilled work.
The paper can take just 15-30 minutes to read, but it may have taken 15 hours to select, synthesize, organize, master, and exposit. When you see one of these that is good, you are deeply appreciative. You learned a trick. You are a better mathematical scientists!
Such papers take skill, wisdom, and knowledge to create. But they do not need more skill, wisdom or knowledge than you have right now. Not all of the papers in this round of Toolbox Essays will hit a high note, but some will. You will be surprised what your colleagues can do. That is the main reason for posting all of the essays.
Hints on How Not to Go Nuts
For beginners, this can seem like a very nebulous task, but it is actually just the tip of the iceberg. In a research project, one has all of these demands plus the stuff you write has to be your own original work --- and it has to have depth. What we relax most here is the depth component; your exposition and insight still have to be original.
It is fine if this is difficult for you. Give it your best shot and see what the other students come up with. You will learn from what everyone does. This is a group experience and everyone contributes to the picture. In that way, it's really like science.
More Example Papers?
This ArXiv paper by Don Knuth has a lovely style. It's longer than our constraint, but if you chop it off at page five, you still have lots of interesting facts. Still, Knuth's style is a very mature style. For beginners, the age-old structure of theorem and proof really does seem to serve best.
At a minimum, the theorem/proof structure enforces mathematical correctness more effectively than a conversational or streaming style. What one might do is follow the theorem/proof format for the first 85% of the paper, and then in a few paragraphs at the end, have a 'discussion" that underscores the new insights that have been found.