Mrthfx

It may be that certain theorems, when proved true, counterintuitively retard
progress in certain domains. Lloyd Trefethen provides two examples:

• Faber's Theorem on polynomial interpolation


• Squire's Theorem on hydrodynamic instability

Trefethen, Lloyd N. "Inverse Yogiisms." Notices of the American Mathematical Society 63, no. 11 (2016).
Also: The Best Writing on Mathematics 2017 6 (2017): 28.
Google books link.

In my own experience, I have witnessed the several negative-result theorems
proved in

Marvin Minsky and Seymour A. Papert.
Perceptrons: An Introduction to Computational Geometry , 1969.
MIT Press.

impede progress in neural-net research for more than a decade.¹

Q. What are other examples of theorems whose (correct) proofs (possibly temporarily)
suppressed research advancement in mathematical subfields?

I don't know the history, but I've heard it said that the realization that higher homotopy groups are abelian lead to people thinking the notion was useless for some time.

I have been told that Thurston's work on foliations (for example: Thurston, W. P., Existence of codimension-one foliations, Ann. of Math. (2) 104 (1976), no. 2, 249–268) essentially ended the subject for some time, even though there was still much work to be done.

Here is a quote from his On Proof and Progress in Mathematics:

"First I will discuss briefly the theory of foliations, which was my first subject, starting when I was a graduate student. (It doesn't matter here whether you know what foliations are.) At that time, foliations had become a big center of attention among geometric topologists, dynamical systems people, and differential geometers. I fairly rapidly proved some dramatic theorems. I proved a classification theorem for foliations, giving a necessary and sufficient condition for a manifold to admit a foliation. I proved a number of other significant theorems. I wrote respectable papers and published at least the most important theorems. It was hard to find the time to write to keep up with what I could prove, and I built up a backlog. An interesting phenomenon occurred. Within a couple of years, a dramatic evacuation of the field started to take place. I heard from a number of mathematicians that they were giving or receiving advice not to go into foliations—they were saying that Thurston was cleaning it out. People told me (not as a complaint, but as a compliment) that I was killing the field. Graduate students stopped studying foliations, and fairly soon, I turned to other interests as well. I do not think that the evacuation occurred because the territory was intellectually exhausted—there were (and still are) many interesting questions that remain and that are probably approachable."

Here I quote from the introduction to "Shelah’s pcf theory and its applications" by Burke and Magidor (https://core.ac.uk/download/pdf/82500424.pdf):

Cardinal arithmetic seems to be one of the central topics of set theory. (We
mean mainly cardinal exponentiation, the other operations being trivial.)
However, the independence results obtained by Cohen’s forcing technique
(especially Easton’s theorem: see below) showed that many of the open problems
in cardinal arithmetic are independent of the axioms of ZFC (Zermelo-Fraenkel
set theory with the axiom of choice). It appeared, in the late sixties, that cardinal arithmetic had become trivial in the sense that any potential theorem seemed to be refutable by the construction of a model of set theory which violated it.

In particular, Easton’s theorem showed that essentially any cardinal
arithmetic ‘behavior’ satisfying some obvious requirements can be realized as the
behavior of the power function at regular cardinals. [...]

The general consensus among set theorists was that the restriction to regular cardinals was due to a weakness in the proof and that a slight improvement in the methods for constructing models would show that, even for powers of singular cardinals, there are no deep theorems provable in ZFC.

They go on to explain how Shelah's pcf theory (and its precursors) in fact show that there are many nontrivial theorems about inequalities of cardinals provable in ZFC.

So arguably the earlier independence results impeded the discovery of these provable inequalities, although I don't know how strongly anyone would argue that.