Project guidelines

1.  Timeline and requirements

The goal of the project is to (1) investigate some problem using the mathematical concepts we’ve studied in this class and (2) write an expository paper on the topic, i.e. explain it in detail to an audience unfamiliar with it.

This is a fairly open-ended project; you may use any resources you like, and the converse of this is that it is your job to find (and properly cite) references to understand your desired topic. That said, I am happy to help you find resources if you are having difficulty, especially on more obscure topics. (You don’t have to notify me about your proposed topic, but if I don’t hear from you I’ll assume you are on top of finding sources etc.)

The project will be due by the end of the day May 1, 2026, the last day of classes.

Guidelines

The primary goal of this project is to understand the mathematics of your topic; nearly as important however is clearly communicating that understanding. Imagine that you are trying to explain the concepts you have studied to someone who has a similar amount of background to you, but has not necessarily studied these particular topics.

Like any paper, in addition to the main body of the exposition your paper should include a short introduction, explaining the main ideas, motivation, and background of your paper, as well as a list of sources. (The specific formatting of your sources does not matter so long as it is clear.)

All papers should be typed. I encourage you to use LaTeX, but it is not required, and you may use whatever software you prefer.

There is no hard guideline for the length of your papers: they should be the length they need to be in order to concisely and clearly explain your topic in detail to the appropriate audience, but in practice I am expecting probably 2-3 pages on the shorter end or 5-7 on the longer end. Precise details of formatting are up to you.

I am open to the idea of projects which do not take the form of a paper, or not solely that form; for example, they could have a coding or graphic component. If you want to do something like this though please discuss it with me first.

Finally, note that as usual in this class the use of generative AI in writing your project is not permitted. However, you may find it useful as a research tool, or to help you understand a topic; if you do use it, make sure to check its sources and statements, as it is particularly good at convincingly justifying claims whether or not they are true (not a skill I want you to pick up!).

Grading

A successful project must study a topic or problem related to complex analysis, but not otherwise contained within the regular subject matter of the course (i.e. the subjects of objectives 1-9). Projects will be graded for mathematical correctness and depth/scope.

Projects will be assigned an overall mark of either S, P, or N. This mark will be your mark for objective 10 (special topics). If your project receives a mark of P, you can raise your mark on objective 10 to an S by correctly answering two questions for that objective on the final exam (to level S); if your project receives an N, you can raise your mark to a P by the same method, but cannot raise it to an S.

In addition, strong projects can earn some number of challenge points: a completely correct paper which solves a very small problem might be worth zero, one or two points, while an exceptionally strong paper perhaps as many as 8-10; any of these should be viewed as a fully successful paper! To earn more points, a project could aim for a larger scope, but note that an ambitious but incorrect paper is less successful than a modest but correct one, and will earn fewer points if any. My expectation is that a project of moderate scope without significant errors should be worth 3-4 points. Some topics are suggested below.

2.  Topic suggestions

Any of these should be taken as a collection of related possible ideas around which to base your project; you do not necessarily need to cover everything mentioned, and might cover aspects not mentioned. Some are closely related to some of our objectives or otherwise covered in Gamelin, others are not.

The prime number theorem.

The prime number theorem gives an estimate for the number of primes less than or equal to a positive real number $x$ (it is approximately $\frac{x}{\log x}$, for $x$ large). This is proven by relating prime numbers to certain complex-analytic functions, and then by integrating along carefully chosen contours one can recover quantities such as the count of prime numbers (or related variants) from the residue formula. Look into this; although the full proof of the prime number theorem may be too big a topic, sketch some of the ideas, and in particular explain where the complex-analytic ideas come in and how they are used.

One version of this story is given in section XIV of Gamelin. Other versions can be found elsewhere. In particular, Gamelin uses a “Tauberian theorem,” which via a trick due to Newman and Zagier makes the proof significantly faster, but arguably obscures some of the complex analysis underlying the argument; you may also want to look at the more traditional argument of Hadamard and de la Vallée Poissin, which gives an explicit formula for a certain prime-counting function using the residue theorem.

Conformal maps.

Conformal maps are functions which “preserve angles” in a certain sense. There are many applications of conformal maps, and relationships with complex analysis. The most fundamental are the relationship between conformal maps and analytic functions (see Gamelin sections II.6, IV.8). Some other applications are listed below; a paper could consist of some basics together with one (or more, in principle) of the below, or your own application.

• Fluid dynamics. Look into the basics of fluid dynamics and applications of complex-analytic methods to problems therein, as discussed for example in sections III.6 and XI.4 in Gamelin. See what other applications you can find elsewhere, and work out some examples.

• The Schwarz lemma. Introduce the Schwarz lemma, as in Gamelin section IX, and study some of its consequences, such as the classification of conformal self-maps of the unit disk and Pick’s lemma. There are also some applications to hyperbolic geometry, as for example in Gamelin section IX.3; you could also more generally explore other features of hyperbolic geometry, its applications and connections to other fields, generalizations, other models, etc.

• The Riemann mapping theorem. In addition to classifying the conformal self-maps of the unit disk, one can also classify the domains in $ℂ$ which map conformally to the disk: it turns out that every simply connected domain other than $ℂ$ itself does. This is one form of the Riemann mapping theorem. Investigate this, and more generally (some of) the content of Gamelin section XI. (To some extent, this builds off of the Schwarz lemma and related ideas.) A full proof of the Riemann mapping theorem might be too big for the paper, but you might be able to give a sketch of the ideas, or describe some related problems and topics.

The Dirichlet problem.

We have studied the problems of finding harmonic functions with certain properties, e.g. harmonic conjugates, and of finding harmonic functions on the disk with given values on its boundary. This is the Dirichlet problem on the disk. More generally, on any domain $D$ we can study the problem of how to extend a given function on $\partial D$ to a harmonic function on $D$; this is the Dirichlet problem on $D$. Research its solution and applications; one version of this story is in section XV of Gamelin, which in particular explains how the Dirichlet problem can be applied to give a proof of the Riemann mapping theorem (see above). There are also many applications and generalizations which you may wish to discuss.

Riemann surfaces.

We have discussed some Riemann surfaces in this class, especially in objective 2, but they are a substantially deeper topic than we have had time to cover. Section XVI of Gamelin gives a more general perspective and proves a uniformization theorem. More generally there is a great deal to be said about Riemann surfaces; say some of it.

The Fourier transform.

For any bounded domain $D$ with piecewise smooth boundary and a smooth function $f:D\cup \partial D\to ℂ$, we can define its “Cauchy integral transform”

$$z\mapsto {\int }_{\partial D}\frac{f(w)}{w-z}𝑑w.$$

Cauchy’s integral formula can then be rephrased as the theorem that if $f$ is analytic on $D$, the Cauchy integral transform of $f$, viewed as a function on $D$, is just $f$ again. The Poisson integral transform and formula give analogues for harmonic functions.

More generally, we could define many other integral transforms. Perhaps the most important of these is the Fourier transform: if $f$ is defined and continuous on an interval $[a,b]$, we can define its Fourier transform on $ℂ$ given by

$$z\mapsto {\int }_a^{b}f(t)e^{-itz}𝑑t.$$

This is defined in Gamelin Exercise IV.6.2. It can also be defined much more generally. Look into some of its definitions, properties, and applications, of which there are many.

Also relevant is the Fourier series, introduced in Gamelin section VI.6. Look into some of its properties and applications, and discuss how it is related to the Fourier transform.

Several complex variables.

The subject we call “complex analysis” could also be referred to as the study of complex functions of a single variable. This phrasing suggests the study of complex functions of more than one variable. Many things can be defined analogously, but some of the behavior is qualitatively different. Describe some of the basic ideas and problems in the study of several complex variables, and discuss the solutions to some of these problems in the one-variable case in terms of results we’ve seen in this class.

Complex geometry.

This can be viewed as a generalization of the previous topic, but in practice is often treated distinctly. For those with some experience with manifolds: recall that a manifold can be described as a topological space which is locally isomorphic to ${ℝ}^n$ for some $n$. A complex manifold should analogously be locally isomorphic to ${ℂ}^n$ for some $n$. But note that differentiable functions on $ℝ$ behave, as we have seen, very differently from differentiable functions on $ℂ$; so requiring our transition functions to be holomorphic makes complex manifolds a different sort of beast from real manifolds; their study is the study of complex geometry. Discuss some of this, and some of the basic ideas and problems in complex geometry, keeping in mind $ℂ$ as the simplest example of a complex manifold.

Choose your own.

Find your own topic! It should be related to the material from this class, so using the methods of complex analysis, holomorphic functions, etc. Otherwise you are free to choose any topic that interests you, using the above as a guide. If you choose your own topic, I suggest discussing it with me so I can warn you if e.g. it seems too hard or not close enough to our class, but this is not required.

3.  Tips

Based on looking through some past comments I’ve given student projects, here are some things to look out for:

• Make sure to define your terms. If defining all the terminology you’re using is especially laborious, consider whether you really need this terminology—do you really use it? If not, how could you cleanly cut it out?

• For technical definitions and arguments that you do include, make sure that they serve the overall thrust of the paper. On the other hand, make sure not to omit so many technical details that the paper loses its technical content!

• Make clear which statements you are proving, which are known but are not proven in your paper, and which may be unknown.

• In terms of writing, don’t overuse symbols: it is generally better to write things out rather than abbreviate. Use symbols only when they are the clearest way to write what you need to, e.g. for formulas.

• Your paper should not read like a collection of bullet points: each paragraph should be coherent. More generally, your paper should not be a collection of facts, but flow naturally, including motivation, results, proofs, and possibly discussions.

• In terms of citations, although the particular citation format doesn’t matter, it should always be clear what the sources of your ideas and material are. For example if you include graphics which you did not generate, these should be cited. If you are mostly using one source, you can indicate this at the beginning and then put citations where other sources are used, there’s no need to be constantly citing the same source every sentence. When you do cite sources, especially long ones, make your citations as precise as possible, e.g. theorem or page number.

• While motivation is important, your paper should not consist entirely of vague discussion: it should have meaningful technical content, whether consisting of proofs, calculations, examples, or some mix. Motivation and discussion should be precise, to the point, and concise.