There is sometimes talk of fields of mathematics being "closed", "ended", or "completed" by a paper or collection of papers. It seems as though this could happen in two ways:
- A total characterisation, where somehow "all of the information" about a field has been uncovered.
- A negative result, rendering the field somehow irrelevant.
A possible example for 1 might be the classification of finite simple groups. Examples for 2 might be Goedel's theorem effectively halting Hilbert's programme, or results showing e.g. certain large cardinal axioms to be inconsistent undermining work which assumes it.
What are some other examples of results "closing" a field?
Are there examples of a small number of papers "completing" a field in the sense of 1 above?
(Apologies for many scare quotes!)
Best Answer
Let me preface this by saying that this is just my own account, based on various conversations I've had over the years with many mathematicians, of the following example.
In 1976, William Thurston proved that a closed smooth manifold has a codimension one foliation if and only if it has zero Euler characteristic. Moreover, every codimension one distribution in the tangent bundle is homotopic to an integrable one.
While history is always more complicated, at least at the folklore level, this result is said to have caused a mass exodus of people working in the theory of foliations. You can read about Thurston's point of view on this, which reflects the history being more complicated, in his note Proof and Progress in Mathematics.
Of course, it's absurd to conclude that this "closed" the theory of foliations. Rather, what I've understood to be the case is that he proved a theorem which was largely expected to be false, and this rendered a nascent industry of building an obstruction theory for co-dimension one foliations largely irrelevant. Nonetheless, I've been told by many people who know way more about this story than I do that graduate students were actively encouraged to avoid the theory of foliations around this time; the general impression being that Thurston was cleaning up the subject.