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I was just reading Timothy Chow's "A beginner's guide to forcing", where I encountered the following interesting concept of an "open exposition problem":
"All mathematicians are familiar with the concept of an open research problem. I propose the less familiar concept of an open exposition problem. Solving an open exposition problem means explaining a mathematical subject in a way that renders it totally perspicuous. Every step should be motivated and clear; ideally, students should feel that they could have arrived at the results themselves."
(Regarding his own paper, the author says that "... it does not solve this open exposition problem, but I believe it is a step in the right direction.")
I would be very interested to hear what others consider to be examples of open exposition problems - so I would like to post this as a question (obviously community-wiki, tagged big-list). I would expect the answers to contain more than just the title of the subject they consider to be the open exposition problem, but also atleast a brief explanation of why they consider this to be so, whether they think this difficulty is inherent in the subject itself, why so, etc. I am aware of the fact that this question might be considered by some to be too subjective/argumentative for MO, and this is why I first ask for your opinions here in meta. My personal opinion is that while this does have the potential to be discussion-y, if the question is properly phrased we could get a list of very interesting, and useful, answers.
So what do you think? Should I post this question? If you think I should, I would also be glad to hear any tips about the shaping of this question before I post it (anything I should add / change / remove). Thanks!
Definitely blog material.
@Andy Putman: You're right. There is definitely the danger of that happening. However, as you mentioned, I can (and will, in case I eventually ask the question) explicitly discourage it. I absolutely do not want this question to be used as a ranting stage for frustrated students to tell us about how hard it was for them to learn a specific subject because the book or papers they read lacked proper motivation or were hard to digest. I would actually want to hear mainly the opinions of experts in their respective fields of expertise, as I feel they are the ones truly eligible to make claims that a certain problem qualifies as an "open exposition problem", and when they make such a claim, you can rest assured it isn't because they have been disappointed with a certain book or resource, but because they have a deep insight into what's happening under the hood in these areas.
For instance, If this question is posted, I hope one of our number theorists suggests "Class Field Theory" as an answer, then tries to explain what makes the (proofs of) class field theory so inaccessible. Coming from an expert, this could definitely be not only interesting, but also insightful and maybe even useful.
@Qiaochu Yuan: Thanks for the link, I have enjoyed reading it. I think MO is a better medium for this question than a blog. For instance in gowers' blog, he has discussed a certain topic, and there have then been 20 comments, but not many "suggestions" (actual list of problems) - well, he hasn't explicitly asked for any. It is my impression that the MO platform has much more potential to draw interesting answers to such a question than a blog post would; for two reasons: 1. The very fact we encourage answers rather than discussions 2. I believe (but may be wrong) that MO has a larger base of mathematicians who would answer this question than blogs have.
I, for one, would appreciate this (in a modified form) as an MO question. I think the topic of "open exposition problem" is a good one: it relates to pedagogy and is something I think many professional mathematicians care about. It is also a common complaint in my (sub-sub-sub-)field, that a lot of new developments that are relatively fundamental yet simple are not presented as such. So such a list can, at the very least, give the senior mathematicians some ideas of what to write for their next Bulletin article. =)
On the other hand, I agree with Andy Putman that such a question (and answers to it), if poorly phrased, can be argumentative and contentious. And I think that the sketch of what you would like to ask, especially the part where you ask the posters to explain why they think current literature is poorly explained, will easily lead to uncomfortable grounds. So at the very least I think you should think a bit more about how you will ask the question in a positive way, and bounce some drafts here on Meta (like GRP did) before you put in publicly on MO.
Regardless of other considerations, this is overly broad! A comparable question would be "Examples of open research problems", and it must be self-evident how vast that topic is.
Thank you all for your comments.
While I would still be very interested in seeing the answers to this question (and I have not yet been convinced that it is not MO appropriate), I respect the opinions expressed here. It seems there are more people who wouldn't like to see this question on MO than there are people who would be interested in seeing it. Therefore I will not be posting the question.
P.S. (added a bit later): this is not so much relevant anymore, since as stated I will not be posting the question anyway, but I just wanted to express my opinion regarding the claims that this question is "overly broad". I don't think that's correct; and I think the comparison of this with a question asking for a list of open problems is not a good one. For two reasons: 1) There are many more "open problems" than "open exposition problems", simply because the latter naturally refers to broad topics and not specific problems (the name "open exposition problems", while catchy, is not accurate, "open exposition topics" would express the concept better, I think); while there may be tens of thousands of well known, previously formulated open problems, I doubt even the most complete list of open exposition problems would contain more than a few hundred items. 2) There are many resources (books, websites) listing open problems, but none that I know of listing, or even discussing, "open exposition problems".
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