Research paper summaries

Background: The following is an assignment I regularly give in my upper-level courses. The task is to find two recent-ish (2000 or later) research articles in a topic related to the relevant course, read them broadly for general content, and then summarize the story that the paper tells. This is geared towards students who have some advanced mathematical experience, but are not in a position to understand every word of any research paper written for professional mathematicians. The point is to learn how to read papers before you know enough to fall into the trap of getting bogged down in the details. Secondarily, reading professional works can help a budding mathematician learn the language of the trade.

The assignment

Find two recent (posted or published 2000 or later) research papers in the general topic of this course, and write a outline/summary of each. The goal is to get a very general understanding of the story each paper is trying to tell, and how it fits into mathematics as a whole; the goal is not to understand every detail of the mathematics it explores or develops.

What to hand in

In a 1–3 page document written in paragraph form, your summary should include to the following:
  • What (sub)field does this research fall under? What is the general context of this research? If it makes connections to areas outside of this field, what are they and how are they connected?
  • What are the major theorems/results? When possible, include a small example to illustrate. What are the authors' favorite examples?
  • If they make a point of it, what are the authors' favorite tools for proving their results or doing calculations?
  • Why is this paper of interest to the general mathematical community? Why is it of interest to you?
Additionally, include information about the context/style of the paper, such as the following:
  • Is there anything about the style of the writing (examples, formatting, etc.) that made this paper particularly easy or difficult to read?
  • Who are the authors? For example, can you find out, when they wrote the paper, what stage of their career were they in? Were they professors? researchers from industry? Where are they based right now? If possible, provide links to their webpages.
  • How/where did you find the paper? If you got it from ArXiV, has it been accepted by a major journal (i.e. peer reviewed) yet?
  • If you work with any of your classmates reading through the same paper, include that information.

Don't get lost

Again, you should not try to become an expert in the paper, nor to understand every definition or detail. The skills this assignment are designed to teach you are (1) skimming papers for highlights, and then maybe looking a little deeper for context and techniques; and (2) learning the language of professional mathematicians and the standard structure of their writing.

A reasonable place to start is in a paper's abstract. But document of yours should read more like an extended abstract, and include specific references to theorem numbering etc. On your first pass through, skip any sections titled "preliminaries". If you need help getting started or processing definitions, you're welcome to come ask for help. But, in general, it's ok to focus on what nouns and adjectives are important to the paper without fully processing what those nouns and adjectives mean.

Choosing papers

Short papers might have fewer details on context and background, and require more of the reader, but long papers might have quite a lot going on. Look for balance. If this is your first time reading research, look for papers with introductions that explain both background and their results in less technical detail than in the body of the paper. For the purposes of this exercise, avoid papers written by undergraduate researchers—some are very good, but the goal of this assignment is to get a feel for professional writing.

Finding papers

  • The ArXiV, especially group theory, representation theory, and rings and algebras, though relevant papers also show up in other sections like combinatorics and quantum algebras.
  • MathSciNet: access this while you're on campus (it's behind a pay wall, but CUNY has a campus-wide license). You can search by MSC code (Mathematics Subject Classification), like 20K01 for "Finite abelian groups" or 05E18 for "Group actions on combinatorial structures". If you try to find a paper via "Find it!" and can't, check the ArXiV or the authors' websites.
  • Of course, there's good old fashioned flipping through journals. Advances, Annals, Inventiones, and Crell are some of math's top-ranked general journals, and Journal of Algebra is the top-ranked journal dedicated specifically to algebra. Each of the major societies, like AMS, LMS, AustMS, etc., have multiple important journals. A relatively complete list of mathematical journals, with their rankings according to AustMS, is here. Again, many of the papers published in these journals are behind pay walls, so either download the PDFs while you're on campus, or look for versions elsewhere once you've found something you want to read.
  • Go to talks, and look up related publications. City's weekly colloquium is on Thursdays. There are also several relevant seminars going on down at the Graduate Center.