The polymath blog

Polymath projects are massively collaborative mathematical research programs, in which a single problem, group of problems, or other mathematical task is worked on by a large group of mathematicians. (While most polymath projects to date have been focused on solving a mathematical problem, one can envision other types of polymath projects in the future, e.g. a collaborative reading seminar, or a collaborative mathematical exposition project, or even some collaborative formulation of conjectures.)

The key word here is collaborative: this is not a competition to be the first to solve the problem, but is instead a team effort, in which each partial insight or other iota of progress gained by any one participant is shared with the other participants via this blog (and also the wiki).

All interested observers are welcome to jump in and participate in any of these projects, regardless of mathematical level, though it is recommended that one read and understand the guidelines here first.

— Proposing a polymath project —

It seems that polymath projects work better if they are announced well in advance, so that the basic format and strategy can be agreed upon before the flood of research observations begins, and so that interested participants can be as prepared as possible for the official opening of the project (which tends to be a bit chaotic).

Some polymath projects are proposed on this blog, but proposals hosted at other places are also certainly welcome. If you are proposing a project on a wordpress blog, please add the category “polymath proposals” so that it can be found on the wordpress page for that category.  If the proposal is not fleshed out enough to merit its own page, it can be placed on the wiki at this location instead.

If you wish to post a proposed project at this blog, please contact one of the administrators of the blog.

An up-to-date list of all active, inactive, and proposed polymath projects can be found on the wiki.

From time to time we may hold a discussion and vote on which proposed project to choose as the next polymath project.  As the polymath enterprise is still very new, we do not intend to have more than one highly active polymath project running at any given time, though we do anticipate a continuing low-intensity stream of activity from previously successful (or partly successful) polymath projects.

— Polymath structure —

A polymath project can be hosted either on this blog, or a blog of a host who is willing to administrate and moderate the project.  A given polymath project consists primarily of three components:

1. The wiki pages for that project. These pages will be hosted on this site, and will store all the “settled knowledge” for that project, such as the bibliography, notation, finished arguments, open problems, and so forth.  Anyone who has registered at that wiki is able to edit these pages.
2. The research thread for the project. This is the engine of progress for the project.  It is in this thread that ideas will be tossed out and analysed, problem solving strategies discussed, and so forth.  Lengthier computations and arguments should be placed on the wiki and summarised on the research thread.  In practice, this thread would be mostly used by the “full-time” participants to the project, who are up to speed on all the existing progress on the problem.  Typically, this thread will accumulate several hundred comments over the lifetime of the project.
3. The discussion thread for the project. This is where the project is managed and evaluated, and where the “casual” participants can catch up and offer feedback.  At regular intervals, the moderators (or other active participants) should summarise how the research thread is going by posting updates to the discussion thread.  Casual participants are encouraged to ask questions or make requests here about the status, direction, or format of the project.  For instance, if the research thread is moving too fast and becoming more chaotic, it might be requested here that more effort go into explaining what has already been achieved, and organising it on the wiki.

Other components to a project may be created as the need arises (e.g. a reading seminar thread to understand a paper of key importance to the project).

Each project will have one or more moderators who can perform various administrative tasks, such as repairing comments with formatting errors, or opening and closing threads.  Participants do not always need to wait for a moderator to act on an administrative issue, though; for instance, if there is a need to give a new update on the discussion thread as to the current status of research, you are certainly welcome to write such a summary yourself.

— Commenting policy —

Polymath projects need comments from participants to move forward.  As such, we welcome all visitors, regardless of mathematical level, to contribute to active polymath projects by commenting on the threads.  However, in order to maximise the positive impact of these comments, the following guidelines should be kept in mind:

1. Be polite and constructive. Polymath projects are team efforts, and any unnecessary conflict between participants is likely to impact that effort negatively.  Criticism of a mathematical argument is welcomed, but personal criticisms should be avoided if possible.
2. Make your comments as easy to understand as possible. If you have an exciting new idea or argument, by all means share it with the rest of us – but please invest a minute or two to try to make your point clear.  For instance, if your argument contains a few messy computations, you might consider writing an initial sentence or two describing the rough strategy of these computations, before plunging into the details; also, please look over these computations and check that they are sensible, and that your terms are well-defined.  If necessary, you might revisit a comment and clarify it in a followup comment (or, in case the comment contains some fixable errors, you might contact a moderator to edit the comment).
3. It’s OK for a mathematical thought to be tentative, incomplete, or even incorrect. Often, progress on a mathematical problem proceeds by first eliminating some ostensibly plausible approaches; the reason for the failure of the approach is often instructive, and gives clues as to what the correct approach actually is.  You are encouraged to toss your ideas out there, but not to be discouraged when others critique that idea.  (On the other hand, please don’t be too sloppy; see point #2.)
4. Excessively technical details should be placed on the wiki, or at another offsite location. If you wish to make a lengthy and very technical post, you might consider putting it on the wiki instead, and posting just a summary of that post here.
5. If you are planning to think about some aspect of the problem offline for an extended length of time, let the rest of us know. A polymath project is supposed to be more than the sum of its individual contributors; the insights that you have are supposed to be shared amongst all of us, not kept in isolation until you have resolved all the difficulties by yourself.  It will undoubtedly be the case, especially in the later stages of a polymath project, that the best way to achieve progress is for one of the participants to do some deep thought or extensive computation away from the blog, but to keep in the spirit of the polymath project, it would be good if you could let us know that you are doing this, and to update us on whatever progress you make (or fail to make).  It may well be that another participant may have a suggestion that could save you some effort.
6. An ideal polymath research comment should represent a “quantum of progress”.  On the one hand, it should contain a non-trivial new insight (which can include negative insights, such as pointing out that a particular approach to the problem has some specific difficulty), but on the other hand it should not be a complex piece of mathematics that the other participants will have trouble absorbing.  (This principle underlies many of the preceding guidelines.)  Basically, once your thought processes reach a point where one could efficiently hand the baton on to another participant, that would be a good time to describe what you’ve just realised on the blog.

Note also that LaTeX is supported in the comments.

— Ratings —

As an experiment, this blog has enabled the ability to rate individual comments “Useful” (Thumbs up) or “Needs Work” (Thumbs down).  The idea is to try to identify the comments in a polymath thread which are particularly useful to the research effort, and also to flag those comments which contain mathematical errors or gaps, or require further clarification.  Ideally, the ratings should assist participants in “catching up” on all the latest developments in a research thread, by identifying the key breakthrough comments, and also noting which arguments turned out to be incomplete or otherwise in need of work.

One should avoid using the rating system to convey personal opinions about other contributors.

— Publishing policy —

If all goes well, a polymath project may end up with one or more publishable results.  If so, the paper will be written collaboratively (using the wiki and blog as appropriate), authored under a polymath pseudonym.  (The question of assessing how much of a contribution each participant had to a project seems impossible to answer with any degree of objectivity, so the pseudonym approach to authorship is the simplest solution.)

In some cases, the results will not be sufficiently conclusive to warrant a formal publication; but a writeup might still be made, to be submitted to a preprint server such as the arXiv, if there is a consensus to do so by the participants.

— Acknowledgements —

The guidelines here are based in part on the guidelines to the first polymath project, the guidelines for the first mini-polymath project, and the comments arising from the subsequent analysis of that project.