Project 2 will involve researching an advanced topic in real analysis that we have not covered yet, as well as an application of that topic. You will be provided with a list of topics to choose from. You will work in groups of three, submitting a portion of your project to the website and also presenting your work to the class.

# Project Guidelines

## Format/Deadlines

The project will have two components:

- Written submission to Wiki site
- Oral presentation to the class

The oral presentations should be completed by **May 13**. Two groups will present on the 13th and one on the 15th. The written portion is due **by 1600 on May 15**.

## Content

Both portions of your project should contain three main components:

- Description of an advanced topic
*directly*related to real analysis (2-3 pages) - Summary of a direct application of the topic to another area of mathematics (1-2 pages)
- Explanation of the relevance of the topic or application (~1 page)

In writing on your topic, you should consider your audience to be your classmates. I will be looking specifically for clarity of explanation, and examples that help to clarify the concepts involved. Part of the project will involve providing feedback on your classmates' work. Here are some further hints on what I am looking for:

- I expect the description of the topic to be detailed. You must include all relevant definitions, and provide examples demonstrating the concept. You should also provide any key theorems, and proofs if they are easily understood.
- I expect the discussion of the application to be somewhat detailed. Primarily, I am looking for explanation of the main question involved, and how it works. Be sure to provide a few examples and figures demonstrating the application.
- As far as the relevance goes, one or two examples suffices. I am looking for something
*beyond*the application that demonstrates where the topic/application can be further applied to solve interesting problems.

Be sure also to include a references (or works cited) section, complete with links if they are available on the web.

## Topics

Possible topics include the following:

Topic | Application | Relevance |
---|---|---|

Series of Functions & Uniform Convergence | Fourier Series | Data Compression (JPEG, MP3, etc.) or Approximating Functions |

Riemann Integration | Fundamental Theorem of Calculus | Integration over Weird Functions |

Riemannian metrics | Non-Euclidean Geometry | Einstein's Relativity Theory |

Calculus of Variations | Generalizing Max/Min Problems | Economic Strategies, Utility of Consumption |

Completion and Compactification | Relationship between n-ball and n-space |
Hyperbolic Space, Mobius Functions, Stereographic Projection |

Curves in Metric Spaces | The Fundamental Group | Classification of Surfaces |

# Guidelines for Mathematical Writing

The following are some tips for good mathematical writing:

- Know your "audience", and keep your audience in mind when 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"… but avoid excessive use of either phrase.
- Keep the passive voice to a minimum.
- Break paragraphs where appropriate; excessively long paragraphs can make a mathematical paper very hard to read.

- Always use 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.

See also http://ems.calumet.purdue.edu/mcss/kevinlee/mathwriting/writingman.pdf and http://edisk.fandm.edu/annalisa.crannell/writing_in_math/guide.pdf for more hints on writing.

# Checklist for Grading (Written Portion)

Primary Topic | Application/Relevance | Technical/Style | |||
---|---|---|---|---|---|

Provides adequate introduction | Application description contains essential points | Adequate voice and tone | |||

Defines any new terms used | Description is clear | Correct grammar/spelling/punctuation | |||

Covers the most important theorems relevant to the topic | Good examples provided for the application | Writing "flows" well | |||

Provides examples demonstrating the topic | Appropriate example chosen to address relevance | Directed towards the appropriate audience | |||

Sketch proofs are given in some form | Clear how the example fits in with the rest of the topic/application | Equations are typeset properly | |||

Mathematics is correct | Describes assumptions where appropriate | Appropriate use of figures | |||

Work appears professional (has been edited, adequate use of wiki syntax, etc.) | |||||

Adequate references given |

I may deem some work worthy of bonus points if something is done really well (e.g. historical background, fantastic proof, great use of figures, etc.)