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Complex Analysis Article Review

This semester you will read a math article about complex analysis and write a review of that article. Your review will count for 10% of your overall course grade. There are many steps to this process and it's best to begin early. Here is a list of the deadlines for the different parts of the review assignment:

Deadlines!

Where can you find suitable articles?

There are MANY places! Mathematics Magazine and the College Math Journal are good places to get started. Use Penfield Library's website to go to the database JSTOR, where you can browse both of these journals and download pdfs of the articles. There are many other journals out there with articles written for teachers, undergraduates, and math aficionado's ... you'll have to do like every other researcher and wade through materials to find the right article for you.

If you've taken a 300-level course with me where a Review Article was assigned then you probably would expect that I'd collect a bunch of potential articles for you here, to save you some legwork. One of the differences between a 300-level course and a 400-level course is that you have to do the legwork yourself. If you want to stop by and get some assistance with the databases and websites, I'm more than happy to chat with you about this – but it's up to you to be proactive this time.

What makes an article suitable?

The goal is for you to learn new things about the complex number system, complex functions, and complex analysis that you wouldn't necessarily see in the classroom and to explain what you've learned to your classmates. Don't try to find an article where you already know everything – after reading your Zeroth Draft, I'll ask you to find a new article anyway! If you find something interesting but aren't sure that it's suitable, show it to me and I'll let you know.

Zeroth Draft

Turn in either a hard copy or an electronic version of the article you intend to review, marked up with the following:

  1. Draw a line in the paper where you no longer understand what's being discussed.
  2. Underline every new word in the paper, even those that are past the "line of understanding".
  3. Highlight or underline the sentences you think are telling you the most important results or ideas of the paper.

You should also include a 1/2 - 1 page plan detailing the steps you will take to learn the new vocabulary and extend your understanding of the article. If the article is too short, too long, too easy, too hard, etc. then I'll ask you to find another article and submit a new Zeroth Draft (no grade penalty).

It's ok to change articles after this point but you will have to submit a new Zeroth Draft for the new article. You cannot change articles any later than 2 weeks before the first draft is due.

Rough Draft and Final Draft

These drafts must be submitted electronically as a PDF, made using latex. You may want to use the latex template provided in the course materials folder, but it is not required that you use the template.

Your review should consist of 1 - 2 pages summarizing the main ideas of the article as well as up to one page response. Throughout the review, keep in mind that your audience is students in Math 347: Analysis A and Math 442: Complex Analysis. By reading this article you have learned all sorts of new mathematics with which your peers are likely unfamiliar. Write in such a way that your colleagues can understand what you're saying!

For the summary portion, you should include:

For the response portion, keep in mind that you do not have to like the writing style of the article – be sure to back up whatever compliments or complaints you make with specific examples from the article. Include suggestions for improvement with specific references to passages in the article.

Tips and reminders for your review:

Grading

Zeroth Draft – All directions should be followed. This will be graded on a +/0/- scale where + indicates that the assignment was completed in a more than satisfactory way, 0 indicates that the assignment was completed as directed, and - indicates that the assignment was not completed in a satisfactory way. In the latter case, the assignment must be turned in again with the missing details added.

Rough Draft – The rough draft will be graded on the same scale as the Zeroth Draft. Not all of the mathematics needs to be perfect and there may be errors in the writing. What matters is that the draft should clearly demonstrate the student's attempt to understand and parse the article, and convey that understanding to others.

Final Draft – The final draft of the review will be graded on the standard letter scale (A/A-/B+/...). Significant improvement should be made even if the improvement was not explicitly directed in the feedback from the first draft. Here, the mathematics, grammar, spelling, etc. must all be correct but also, the ideas of the article should be made clear to the reader.

Project

The complex number system, complex functions, and complex analysis are woven into every aspect of mathematics and have tendrils spread out into fields such as physics, computer graphics, and computer science. This semester you will complete a project that explores the depths of the complex field, beyond what we have time to study in this class. There is no "one size fits all" project. You'll have to follow your interests and build a project that fits you.

Deadlines!

Proposal

You must submit a project proposal, to be approved by me, before anything else on the project can be graded. The proposal must describe the topic, list the individuals working on the project, outline the goals for the project, and contain a list of potential references (properly cited, of course). The project will culminate in a presentation during the final exam time period and a write-up, but there can be other "products" – a solver, a program, a sculpture, ... The proposal should describe these "products" and demonstrate that the project is deep enough to merit 15% of your course grade and requires the number of individuals on your team.

While Wikipedia is a good place to get started, , the important thing to remember is that you cannot stop at this website. You MUST go to the sources cited by these resources and get the information from the original sources or sources CLOSE to the original sources. Keep track of all of your resources so that you can properly cite them on the poster. Even figures and images must be cited, even if they are from Creative Commons.

Progress Report

The progress report should describe what progress has been made towards meeting the goals for the project, discuss any challenges or highlights thus far, and address any changes that might need to occur to the plan laid forth in the proposal.

Write-Up

Different projects will call for different requirements in the write-up. Some write-ups will be 2-3 pages, but others could be 10-12 pages -- it really depends on the project and what the authors are trying to create with the project. All writing should use complete sentences, correct grammar and punctuation, and use good mathematical style. The formatting guidelines for the review article apply here, too. This is especially true for the citations, which should be peer-reviewed (where possible) and trusted sources.

Presentation

We will divide the 2-hour final exam time period into fair shares for the presentations. You should dress neatly, practice and time your presentations, and behave formally. Treat the presentation like a practice for a capstone presentation or another presentation in your college career (or beyond!). You are likely to need to prepare slides, which should be proof-read, carefully designed, and appear tidy.

A Few Ideas...

  1. investigate the Riemann zeta function.
  2. investigate complex dynamics, chaos, fractals, and the Mandelbrot set (Check out: Explorations in Complex Analysis by Michael A. Brilleslyper, Michael J. Dorff, Jane M. McDougall, James S. Rolf, Lisbeth E. Schaubroeck, Richard L. Stankewitz, and Kenneth Stephenson).
  3. study applications of the Schwartz-Christoffel Transformation (See Complex Variables and Applications by Ward and Churchill).
  4. study the connections between complex analysis and signals processing (See the article "Connections between signal processing and complex analysis " by J. Bruna, X. Massaneda, and J. Ortega-Cerda for starters).
  5. investigate using the complex number system as a background setting in which to teach euclidean and non-euclidean geometry at the high school level.
  6. study models of hyperbolic geometry embedded in the complex plane.
  7. study models of elliptic geometry embedded in the complex plane.
  8. discover the connections between the complex plane and computer graphics, by way of conformal mapping.
  9. study the lives and works of famous mathematicians who dived deep into the development of complex analysis.
  10. study quantum computing models, which make use of complex numbers and matrix algebra.

References

You may use the internet to help you get started. Websites such as Wikipedia articles often can give pretty good explanations. You must, however, turn to peer-reviewed sources for additional information. Look in the references listed at the bottom of a Wikipedia article. Look for textbooks and journal articles. There are books entirely about unsolved problems and famous conjectures! Student class papers posted on the internet do not count as peer-reviewed sources.

Be sure to cite your sources in a proper format. Acceptable formats include MLA, APA, and the bibtex plain style.

Writing Tips

Writing mathematics well is not as easy as it first seems. Many established mathematicians are poor writers, but even good writers struggle to improve their writing skills every time they write. You can read more about the expectations of good writing in mathematics by checking out this article: A Guide to Writing Mathematics by Dr. Kevin P. Lee. We'll talk about my specific expectations as we work through the semester, but please know that you are always welcome to ask and you will be given options to rewrite and improve your work. We all have room to improve our writing!

Your Writing Assignment grade will be based on (1) typesetting and (2) good writing skills. There will be a separate grade for mathematical correctness. Sometimes people write well but don't actually do the assignment ... other times people can get the right answer but explain it poorly. That's why the grades are separated. Questions? Come talk with me!


Getting Started with LaTeX

There are many online and free ways to access latex, including Overleaf, CoCalc, LatexBase, Papeeria, Authorea, ... Each one is different. There are also the editors/compilers that you can download and install on your own machine (such as MiKTeX and MacTeX for compilers, and editors like TeXworks). Stop by office hours if you want some assistance in getting started.

Latex Figures and References

The article review template has examples of one way to do references in latex. The template requires that you type in all of the bibliographic information for each reference item (bibitem) individually and format the entry yourself. It's not that bad if you have only one source to cite. On the other hand, if you are citing several sources (either in the review article or the poster) then you may be interested in learning how to use bibtex. People love bibtex. The formatting is automatic! It's so straightforward!

To refer to a bibitem in your latex document, use the command \cite{key}. Each source has its own identifier, or key. When you type \cite{key} then the after a few compiles, the compiler realizes where the bibliographic info is, how many sources there are, numbers them, and then a little [#] will appear in your pdf with the correct number. The simplicity and automation are heavenly.

To get a figure into latex, first make sure the picture file is a png or jpg. Other types of files will work, but the coding is a bit more complicated. Screenshots tend to be png files so if your file is not a jpg or png already, we can get your figure into a workable format in no time. Then make sure the picture file is in the same folder as your .tex file. Use this code for pngs:

  
    \begin{figure}[H]
    \centering
    \caption{Put your caption here.}\label{key}
    \includegraphics[scale=1]{filename}
    \end{figure}

Replace "filename'' with the actual name of your picture file. For instance, if the picture is tulips.png then type in "tulips". Also change the work "key" to something that uniquely identifies this figure, for instance "tulips''. When you refer to the figure, you can type "Figure~\ref{key}" and the number will automatically show up. It's awesome! If the png is too large or small, you can change the number that the scale is set at as appropriate.

For a jpg use the code:

  
    \begin{figure}[H]
    \centering
    \caption{Put your caption here.}\label{key}
    \includegraphics[width=4in]{filename}
    \end{figure}

Here, you need to set the width of the picture to the appropriate number of inches. You can use centimeters, instead, if you prefer.

Questions? Confused? Worried? Well, come to office hours or send me your .tex file!!

LaTeX Fun

You can exchange the commands in the box below for your own and render it on demand. It's kind of fun! Press the Render! button when you're ready.

Let’s roll that beautiful math footage!

\({}\)