I have an idea for a website that could improve some well-known difficulties around peer review system and "hidden knowledge" in mathematics. It seems like a low hanging fruit that many people must've thought about before. My question is two-fold:
Has someone already tried this? If not, who in the mathematical community might be interested in creating and maintaining such a project or is working on similar projects?
A website dedicated to anonymous discussions of mathematical papers by experts.
Motivation 1: Hidden knowledge
Wilhelm Klingenberg's "Lectures on closed geodesics" can be found in every university's math library. One of the main theorems in the book is the following remarkable result, a culmination of decades of work on Morse theory of the loop space by many mathematicians: Every compact Riemannian manifold contains infinitely many prime closed geodesics.
Unfortunately, there is a mistake in the proof. 44 years after the book's publication the statement is still a widely open problem. The reason I know this is because when I was in grad school I mentioned the book to my adviser and my adviser told me about it. If I tried to look for this information online I wouldn't find it (in fact, I still haven't seen it written down anywhere).
This is one of many examples of "hidden knowledge", information that gets passed from adviser to student, but is inaccessible to an outsider. In principle, a new Ramanujan can open arxiv.org and get access to the cutting edge mathematical research. In reality, the hidden knowledge keeps many mathematical fields impenetrable to anyone who is not personally acquainted with one of a handful of experts.
Of course, there is much more to hidden knowledge than "this paper form 40 years ago actually contains a gap". But I feel that the experts' "oral tradition" on papers in the field is at the core of it. Making it common knowledge will be of great benefit to students, mathematicians from smaller universities, those outside of North America and Europe, people from adjacent fields, to the experts themselves and to the mathematical progress.
Motivation 2: Improving peer review
Consider the following situations:
- You are refereeing a paper and get stuck on some minor issue. It will take the author 5 minutes to explain, but a few hours for you to figure it out on your own. But it doesn't quite feel worth initiating formal communication with the author through the editor over this and you don't want to break the veil of anonymity by contacting the author directly.
- You are being asked to referee a paper, but don't have time to referee the whole paper. On the other hand, there is a part of it that is really interesting to you. Telling the editor "yes, but I will only referee Lemma 5.3" seems awkward.
- You are refereeing a paper that is at the intersection of your field and a different field. You would like to discuss it with an expert in the other field to make sure you are not missing anything, but don't know a colleague in that area or feel hesitant revealing that you are a referee for this paper.
These are some of many situations where ability to anonymously discuss a paper with the author and other experts in a forum-like space would be helpful in the refereeing process. But also outside of peer review mathematicians constantly find small errors, fillable gaps and ways to make an old paper more understandable that they would be happy to share with the others. At the same time, they often don't have time to produce carefully polished notes that they would feel comfortable posting on arxiv, or if they post notes on their website they may not be easy to find for anyone else reading the paper. It would be helpful to have one place where such information is collected.
How will it work?
The hope is to continue the glorious tradition of collaborative anonymous mathematics. One implementation can work like this: Users of the website can create a page dedicated to a paper and post questions and comments about the paper on that page. To register on the website one needs to fill in a form asking for an email and two links to math arxiv papers that have the same email in them (this way registration does not require verification by moderators) and choose their fields of expertise. When a user makes a comment or question only their field/fields are displayed.