This is a soft question but it's not meant as a big-list question. I have recently been asked whether I want to provide feedback at the pre-beta stage on a forthcoming website that will provide a platform for data sharing, and rather than giving just my personal opinion I'd rather consult other mathematicians first. I was going to write a blog post but then I thought that Mathoverflow was a more suitable place since I have a question and I'm looking for answers of a certain type rather than general comments. The website seems to be aimed mostly at scientists who want to share raw data, so at first I thought it probably wouldn't be much use to mathematicians since our data is (or are if you prefer) mostly highly interlinked — the connections are often more interesting than what they connect.
But on further reflection, it seems to me that a good data sharing site could be a valuable resource, even if it doesn't do absolutely everything any mathematician would ever want. For instance, Sloane's database is fantastically useful. A rather different sort of database that is also useful is Scott Aaronson's Complexity Zoo. So useful databases exist already. Is this an aspect of mathematical life that could be greatly expanded given the right platform? And if so, what should the platform be like?
I don't know anything about the design of the site, but if I'm going to comment intelligently on what features it would need to have to be useful to mathematicians, I'd like to be armed with some examples of the kind of data sharing we might actually go in for. Here are a few ideas off the top of my head.
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Diophantine equations: one could have a list of what is known about various different ones.
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Mathematical problems: listed in some nice categorized way, each problem accompanied by a description, complete with reading list, of what you really ought to know before thinking about the problem. (As an example, if you are thinking about the P versus NP problem, then you really ought to know about the Razborov/Rudich natural proofs paper.)
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Key examples in various different areas and subareas of mathematics.
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Sometimes you have a whole lot of related mathematical properties with a complicated pattern of implications between them. Under such circumstances, it could be nice to have this information presented in a nice graphical way (something I think this site may be able to do well — they seem to be keen on visualization) with links to proofs of the implications or counterexamples that demonstrate when the implications do not hold. (The example I'm thinking of while writing this is different forms of the approximation property for Banach spaces, but there are presumably several others.)
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List of special functions and the facts about each one that are the main facts one uses to prove things about them.
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List of integrals that can be evaluated, with descriptions of how they can be evaluated.
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List of important irrational numbers with their decimal expansions to vast numbers of places. (I'm not sure why this would be useful but it might be amusing.)
These are supposed to be examples where people could usefully pool the background knowledge that they pick up while doing research. I'm not particularly pleased with them: they should be thought of as a challenge to come up with better ones, which almost certainly exist. If you've ever thought, "Wouldn't it be nice if there's somewhere where I could look up X," then X would make a great answer. I think the most interesting answers would be research-level answers (unlike some of the suggestions above).
If there were a site with a lot of databases, it would make a great place to browse: it would be much easier to find useful data there than if it was scattered all round the internet.
One constraint on answers: there should be something about a suggested database that makes it unsuitable for Wikipedia, since otherwise putting it on Wikipedia would appear to be more sensible.
Best Answer
It would be really useful to have a database containing various computations of (co)homology/homotopy groups of various spaces that arise in algebraic topology...
Note: There is so much known out there that one would have to first think really hard about how to organize it all.
Here's an example:
I could imagine that, for certain users, listing the first 30 integral cohomology groups of the spaces $K(\mathbb Z,1)$, $K(\mathbb Z,2)$, $K(\mathbb Z,3)$, and $K(\mathbb Z,4)$ could be more useful¹ than listing all the cohomology groups of all the $K(\mathbb Z,n)$'s. The reason is that, in order to do the latter, the information has to be packaged in a certain way that might be hard to understand: the user would need to unpack that information before she can access it.
¹ Of course, it's even better to have both pieces of information available.