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.
- Compactness — A compact set in a metric space has the Heine-Borel property, that every open cover has a finite subcover. Compact sets have a number of important properties and applications, including a generalization of the Extreme Value Theorem to metric spaces.
- Connectedness — The definition of a "connected" set in a metric space is not very intuitive, but it has interesting and important consequences, including a generalization of the Intermediate Value Theorem to metric spaces.
- Completeness — Completeness in metric spaces is a generalization of the completeness of the real numbers, but it is defined in terms of Cauchy sequences rather than the least upper bound axiom. Every metric space has a "completion," a complete metric space in which it is embedded as a dense subset.
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.
- Fourier Series — This course will consider power series, which are like infinite polynomials. A Fourier series is an infinite sum of sine and cosine functions. Almost any reasonable periodic function can be written as a Fourier series.
- Complex Analysis — Analysis using the complex numbers instead of the real numbers has a number of surprising properties, such as the fact that any differentiable function is infinitely differentiable. Chapter 7 in our textbook is an introduction to complex analysis.
- Topology — We have already encountered topological spaces. It might be interesting to look at some examples that are not metric spaces, or to see how metric space properties extend to topological spaces.
- Fractals — It's hard to say exactly what a fractal is, but the term is often used to refer to "self-similar" sets or to sets that have a "fractional dimension." The Cantor set, for example is a self-similar fractal with fractal dimension ln(2)/ln(3).
Examples. Here are a few specific examples related to things we have covered. I might do some of them if no one else does.
- A Non-Archimedian Ordered Field — Rational functions can be used to construct a field that contains the real numbers as a subfield but is non-Archimedian and therefore contains elements that are greater than every integer. This example was covered in a reading guide section from a previous version of this course.
- The Cantor Function — a function that is continuous and strictly increasing but has derivative zero almost everywhere. It is based on the Cantor set. (For the meaning of "almost everywhere," see below.)
- Rearrangement of a Series — A conditionally convergent series can be rearranged so that the rearranged series converges to any given real number.
- An Infinitely Flat Function — the function f(x) which is zero for x less than or equal to zero and is equal to e-1/x2 for x greater than zero. This function is infinitely differentiable and all of its derivatives at zero are zero. It can be used to construct other interesting examples.
Measure Theory. "Measure" is a generalization of concepts of size, such as length, area, and volume, and it is important in advanced mathematics.
- Sets of Measure Zero — Sets whose measure is zero are particularly important, and they can be defined without knowing anything about Lebesgue measure or measure theory in general. A set can have measure zero and still be large in some sense. For example any countably infinite set has measure zero. And the Cantor set is an example of an uncountable set that has measure zero.
- Lebesgue Measure — Lebesgue measure is the usual measure on the real numbers. The Lebesgue measure of an interval is its length. But there are sets that are not Lebesgue measurable. (The existence of such sets depends on the Axiom of Choice.) It is also possible to talk about measure in general.
- The Lebesgue Integral — The Lebesgue integral is a generalization of the Riemann integral that uses Lebesgue measure. Instead of partitioning an interval into subintervals, it partitions it into Lebesgue measureable sets. For various reasons, it is the preferred integral in much of advanced analysis.
- The Banach-Tarski Paradox — A sphere can be disassembled into five pieces which can then be reassembled into two complete spheres of the same size. But the pieces are unmeasureable sets.
- Riemann Integrability — There is a complete characterization of Riemann integrable functions: A bounded function on [a,b] is Riemann integrable if and only if it is continuous except on a set of measure zero. There are examples of Riemann integrable functions that are discontinuous on an uncountable set of measure zero, such as the Cantor set.