Project I Description

An essential skill in the mathematics profession is the ability to read an article and understand the key points in the article. The first project (50 points) will be to read an article in a mathematics journal and to write a summary of the article.



Your summary should be a few pages long. The appropriate audience for your summary is an undergraduate student who is interested in mathematics. Here are the things I will be looking for:

  • Content
    • Convey the main point of the article, why the question at hand is interesting, and the essential ideas of the proof.
    • Describe the connection between real analysis and the article.
    • Provide context for the article. What is the broader field that looks at questions like these?
    • Mention how the interested reader could study more results like these.
  • Writing style/correctness
    • Proper spelling, grammar, etc.
    • Good organization and flow
    • Proper mathematical and wiki formatting

Format of Submission

The summary should be posted to the course wiki, and be formatted using proper Wiki syntax. To ensure that you do not lose your work, you should either save frequently or copy/paste what you write offline into the edit box.

To begin your article, type into the address bar "" (replacing last-first with your name). Click "create page", give it an appropriate title, and next to "Initial Template" select "Project I Template". This will load up a template which you may use.

Choosing an article

You should choose an article of appropriate length and content. The article needs to be related to real analysis in some sense (e.g. limits, functions/continuity, etc.), and should involve a proof of some sort. It may be drawn from a "Notes" section if it involves a proof directly related to the material we are currently studying.

Guidelines for Mathematical Writing

The following are some tips for good mathematical writing:

  • Never use the first person singular ("I"); both third person and first person singular ("we") are frequently used in mathematical writing.
  • You may refer to "the author" or "the reader" but not "me" or "you".
  • Keep the passive voice to a minimum.
  • Break paragraphs where appropriate; excessively long paragraphs can make a mathematical paper very hard to read.
  • Know your audience, and keep your audience in mind when writing.
  • Always used complete sentences, including when mathematical equations are involved.
  • Be explicit when citing a theorem or giving a definition; remember that mathematics is a very precise language.
  • Define terms before using them.

Checklist for Grading

Content (30 points)

$\square$ Describes/understands the main ideas of the articles
$\square$ Includes main theorem of the author
$\square$ Gives adequate introduction for the uninitiated reader
$\square$ Speaks to the work involved in proving the main result

$\square$ Adequately describes a connection between real analysis and the article (e.g. limits, continuity, etc.)
$\square$ Gives reader a feel for the broader context of the result
$\square$ Provides appropriate further references

Writing/Style (20 points)

$\square$ First person is not used
$\square$ Grammar/spelling/punctuation are correct
$\square$ No excessively long paragraphs or sentences
$\square$ Writing "flows" well
$\square$ Written at an appropriate level (audience)

$\square$ Terms are defined as necessary
$\square$ Theorems are stated with adequate qualifiers
$\square$ Equations are typeset properly

Links to Projects

  • Addison Bohannon: The Rationals of the Cantor Set by Ioana Mihaila in The College Mathematics Journal
  • Paul Falcone: The Changing Concept of Change:The Derivative from Fermat to Weierstrass: Judith V. Grabiner in Mathematics Magazine
  • Chris Grauel: Rental Harmony: Sperner's Lemma in Fair Division by Francis Edward Su in American Mathematical Monthly
  • Bryan Jonas: Six Ways to Sum A Series by Dan Kalman in The College Mathematics Journal
  • James Lee: The Possiblity of Impossible Pyramids by Tom Sibley in Mathematics Magazine
  • James Morgan: The Rationals Are Countable: Euclid's Proof by Jery Czyz and William Self in The College Mathematics Journal
  • Kyle Morgan: Advanced Plane Topology from an Elementary Standpoint by Donald E. Sanderson in Mathematics Magazine
  • Geoff Phillips: The Golden Section and the Piano Sonatas of Mozart by John F. Putz in Mathematics Magazine
  • Dominic Senteno: Compact and Connected Spaces by Richard Johnsonbaugh in Mathematics Magazine
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