Math 331, Foundations of Analysis, Fall 2022
Final Project


This course requires a final project, which takes the place of a final exam. The project couts for 13% of the course grade, the same as the in-class and take-home tests.

For your project, you will write a short paper and do a presentation about a topic chosen in consultation with me. All the topics must be different, so if you are interested in some particular topic, you should make a claim on it early. You should select a topic by November 11.

It is possible that a group of two or more people will work together on a large topic or on several related topics. However, each person in the group should submit a paper and do all or part of a presentation.

Most of the presentations will take place during the scheduled final exam period for this course: Wednesday, December 7, 7:00 to 10:00 PM. However, since it would be difficult to fit all eleven presentations into that time period, some should be given during the last week of classes.

Your paper should be about three or four pages long, and it should give an overview of the basic mathematical ideas related to your topic. It will probably include definitions and statements of theorems, along with a more informal or intuitive discussion. It will not necessarily include proofs, but you can include them if it seems appropriate. Papers are due at the final exam period.

You can give either a chalkboard presentation or one using something like PowerPoint. It should be approximately fifteen to twenty minutes long.

Topic Ideas

Here are some possible topics for you to consider. You are not restricted to choosing a topic from this list. If you have other ideas, discuss them with me!

Metric Spaces. We have covered only a few of the basic concepts from the theory of metric spaces, such as open and closed sets, continuity, and sequences. There are several additional topics that I would like the class to encounter. I will probably cover these topics myself if no one in the class does them. Each of these topics has a section in my online Introduction to Metric Spaces that you can consult.

Beyond Foundations. There are many extensions and applications of basic analysis. Metric spaces are just one example. These topics are too broad for you to cover in any depth, but you might be able to give an overview of some of the definitions, results, or applications.

Examples. Here are a few specific examples related to things we have covered. I might do some of them if no one else does.

Measure Theory. "Measure" is a generalization of concepts of size, such as length, area, and volume, and it is important in advanced mathematics.