From what I’ve gathered through talking to people on their way to a Ph.D., there’s quite a difference between the mentality required for coursework and the mentality required for research. For instance, in a course, the subject matter has been distilled and organized in a particular manner by the instructor. There is a clear path to follow: attend the lectures, read the lecture notes, read the text books and other references, solve problems, do the assignments, and sit the examinations. Even if everybody does not follow this clear path, the fact that it exists is a source of reassurance.. it is always there to fall back upon.

Doing research, which may involve solving open problems or extending existing results, is a different ballgame. At best, the student is given a problem and some material to chew upon and is then practically let loose on it. At worst, the student is told to find his or her own problem and work on it and keep using the advisor for course correction. Clearly, a completely different approach is required for this: an approach where the student figures out what information to collect, how to collect it, how to use it, whether to discard it, and so on.

An important distinguishing feature about research orientation, then, is broad reading with a narrow focus and a specific objective. Since this kind of focus cannot be provided in the routine college environment, students keen on developing the research orientation need to find other means of developing the skills. Summer schools and summer camps usually help in providing such focus. For instance, in the VSRP programme at TIFR, that I attended this summer, I was asked to read a paper on Lie Group Representations of Polynomial Rings. Here’s a link to the final presentation I gave on a part of the paper. I’ve chronicled about this paper in earlier posts on this same blog. Check out this post and subsequent posts.

Students can be encouraged to read papers even within coursework, by having student seminars as part of the course accreditation. Some of my courses at CMI this semester have student seminars. For instance, in the course on Representation Theory of Finite Groups, each student is supposed t give a seminar on a topic decided by the instructor; I have to give my seminar on Artin’s Theorem. In the Elementary Differential Geometry course (course details are available here), a list of seminar topics was given and each student had to select a topic, I chose the Whitney embedding theorem and a write-up of what I presented is available here.

Apart from reading research papers for these courses, I have also been reading research papers to seek and collect knowledge in various areas of mathematics.

First, some differences between the textbook and the research paper:

- The textbook presentation, or the lecture note, is meant to be an introduction to the subject. It is intended to provide overall motivations, basic definitions, and a level of familiarity and comfort to people who are new to the subject. Steps are left out or missed only if they are easy for the reader to fill out or filling them is an instructive exercise for the reader.

The research paper, on the other hand, is meant to be a concise introduction to a new discovery or a new idea or a new formulation, for people who are already familiar with the area. Definitions and background are provided only in order to set notations and conventions, explain the authors’ mindset and revive the memory of readers. Efforts are not made to be complete. Further, the authors tend to skip on steps which: (a) have been proved elsewhere (b) require routine checking that other experts can do (c) provide no insights and detract from the essence of the paper. - A (well-written) research paper has a clear end in mind, which it tries to outline in the beginning. It then gradually builds up the arsenal and ammunition needed towards proving this end. At some point in the paper, the authors usually discuss how this new result sheds new light in the areas being explored.

A textbook, on the other hand, may not have a clear, specific result that it intends to establish. Rather, it aims to develop a backdrop and a framework in the minds of students.

Based on my experiences (both positive and negative) in trying to grasp research papers, I have come up with the following strategy:

- Try to get an idea of what the paper is trying to prove. This can usually be gleaned from the abstract, from the introduction, or from the beginning of the second section (if the first section is for preliminaries).

Look for something marked Theorem 1 or Main Theorem. - Understand carefully the statements of previously written results in that area, and use that understanding to try to figure the import of the new result obtained. Try to state the new result obtained in as many different flavours as possible. Make all of them as appetizing as can be!
- Now, look at the statements of the lemmas and corollaries, and try to understand each statement. Attempt a broad trajectory that describes how the theorem is obtained, via the lemmas and corollaries. Do not look at the proofs yet, unless they help significantly in understanding the statements.

While trying to understand the statements of the lemmas and corollaries, it may be necessary to familiarize oneself with the notation of the paper. - After a short break, look at this trajectory, and try to figure out which steps in the deduction process are clear and obvious. Often it may happen that many steps in the deduction process are not too hard. Figuring out that one already understands a lot of the proof before having seen the actual proof is a great confidence-booster.

For the parts where the proof seems clear, look at the actual proofs and see whether they match the proof in your mind. - Now, it is time to focus on the non-obvious parts of the proof. Gently look at the proofs of each of these. Some of these may turn out to be clear once you read the proof. For others, however, the proof may involve some new idea. Zero in on the proofs that are hard to understand. Note the crucial leaps of thought. Don’t be in a hurry to digest these pieces.
- Come back after another break. Recall the proof skeleton, and the proofs of the easy part. Now, in easy sessions, master the hard parts. Take special care to master those parts that fill you with the maximum discomfort.

This approach steadily zooms in on the proof details by beginning at the main result, then proceeding to the proof skeleton, and then finally going to the nitty-gritties of the actual proof.

What are the kind of results one obtains with this approach?

Some observations:

- Steady documentation at each step is particularly useful. In this respect, I think one way of documentation is to prepare a presentation on the paper. A nice tool for preparing presentations is the document class beamer in LaTeX.

Here’s an example of the PDFized version of a file using beamer: An electric story of a drunkard. The original LaTeX file looks like this. - Often, reading a research paper is disconcerting because one realizes the many gaps in one’s knowledge on encountering statements that the authors claim are obvious but that are not obvious at all. This has happened to me quite often! But whenever I have followed this zoom-in strategy of first concentrating on the broad motivations, then strengthening the proof skeleton, and then going in for the actual proof details, I have found that the disconcerting parts only come towards the end, by which time I have already gained a lot of confidence in the paper.
- The “zoom-in approach” works best if the reader is used to looking at things and ideas in terms of their motivations, and understands the broad motivations in the topic where the research paper was written. These motivations are meant to be developed in the regular coursework, through comments and remarks made by the instructor, through the structure of the course outline, trough comments in the book, through the choice of exercises and problems that the student solves.

However, even students not used to looking at things motivationally can start doing so by applying the zoom-in approach to a given paper!

Now, a chronicle, of some of the mistakes I have made when reading research papers:

- Reading the first two pages and then quitting: True, this isn’t a really bad thing if the paper is well-written, because the author would have put the statement of the main theorems in the first two pages. However, simply knowing the statement of the theorem, without understanding the proof skeleton, may sometimes be useless.

In some cases, the proofs may be hard. But in the past, I have often skipped the proofs simply because they seemed too tedious. However, now that I have started applying the “zoom-in” approach, I am able to absorb a little of the proof skeleton even if the steps of the actual proof remain unclear. - Getting disheartened because many statements in the beginning don’t seem to make sense: The introduction of the research paper usually contains both background preliminaries and a summary of important results shown in the paper. While reading the paper on Lie Group Representations of Polynomial Rings, I thought that the first few pages contained background preliminaries, and was disheartened at the fact that figuring out their meaning took me a lot of time. Only after crossing those initial pages did I discover that the content of the first few pages was not background preliminaries, but results proved in the paper.

To avoid confusing background preliminaries (viz what is assumed) and the core content of the paper (viz what is established/proved) it is important to have a look at the whole paper. A strategy that I have followed since the experience with the Lie Group Representations paper is to create a mapping of the introductory section onto the rest of the paper. This way, it is clear to me which parts of the introduction have what purpose. - Not having any clear targets: A huge research paper can be daunting, but at the same time, it may be difficult to set intermediate targets. That’s what happened with the Lie Group Representations of polynomial Rings paper. It took me a lot of time to get a hang of the structure of the paper.

In retrospect, I feel that after mapping the paper, and getting a hang of its structure, I should have singled out the results that it was important for me to master, and then applied the “zoom-in” approach towards mastering them.

I’ll post more on this. Looking forward to comments in the meantime.