In mathematics, it is common that theorems/results and problems appearing dull in one generation get revitalized and become the center of research in another one.
Sometimes conjectures that are thought to be untouchable become resolved within years.
I would like to know if there were conjectures that did not appeal to a large number of mathematicians but when they were resolved( maybe partially), the techniques used or the result itself touched upon many areas.
On the other hand, are there also conjectures which were expected to have dramatic impact but upon resolution did not meet to that expectation?
Best Answer
An example of an important solution to a little-known problem might be Frank P. Ramsey's "On a problem of formal logic" in Proc. London Math. Soc. 30 (1930) 264-286. The problem was in logic and not well-known even to logicians, but Ramsey's solution was taken up by combinatorialists (notably Erdős and Szekeres) and it grew into the important field now known as Ramsey theory.
{Added later] An example of the contrary type is Hilbert's fifth problem. This was a well known and difficult problem, worked on by eminent mathematicians such as von Neumann and Pontryagin, and it took more than 50 years to solve. Yet, by the time it was solved it seemed to be no longer in the mainstream of Lie theory, and books on Lie theory today make little mention of it.
PS. I agree that this question should be community wiki.