# What are examples of (collections of) papers which “close” a field?

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:

1. A total characterisation, where somehow "all of the information" about a field has been uncovered.
2. 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!)

• I'm not sure that I'd consider the classification of finite simple groups to be closed; there's still a lot of work on trying to simplify the proof, or to prove some results without relying on the full classification. – JoshuaZ Dec 3 '19 at 14:54
• As one poet used to say, "yesterday I wrote four poems about love. Closed the subject." – Fedor Petrov Dec 3 '19 at 20:57
• An upper bound on the year of publication would be a good safeguard against too opinion-based answers. – YCor Dec 3 '19 at 22:35
• I would guess Markov Jr proof that the classification of four-dimensional manifolds is undecidable possible made some people change subjects but I don't think it closed a field to classify as an answer per se, so just leaving it as a comment – Daniel D. Dec 5 '19 at 18:08
• Since it's a bit late to post an answer, I leave this as a comment. For two millennia, mathematicians tried to prove the parallel postulate. But in the 19th century, the field was effectively closed and sealed by introducing non-Euclidean geometries ―thanks to Lobachevsky, János Bolyai and Riemann. The independence of the parallel postulate from Euclid's other axioms was finally demonstrated by Eugenio Beltrami in 1868. – polfosol Sep 13 '20 at 16:29

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.

• As an important side note, to whatever extent Thurston can be said to have "closed" the theory of foliations, it is exponentially eclipsed by his "opening" of the theory of three-dimensional geometry and topology: this latter phenomenon would serve as a nice counterpoint to the asked question. – Andy Sanders Dec 3 '19 at 15:43

In this classic article, Steinitz closed not just one, but all fields.

• That's horrible, +1. – Noah Schweber Dec 4 '19 at 19:35
• Can you elaborate? This answer has 47 upvotes, but I have no idea why. – Fabian Röling Dec 5 '19 at 13:29
• @FabianRöling it's a play on words that perhaps only makes sense in English. The original question asks for papers which "closed fields", which in context means papers which essentially settled all pertinent questions in a mathematical area, leading to said area's death. However, 'field' in English also means a mathematical field (i.e., $\mathbb{R}, \mathbb{C}$ etc), and the cited paper proved that all fields have a unique algebraic closure up to isomorphism, hence 'closing' all fields. – Stanley Yao Xiao Dec 5 '19 at 18:05
• This is funny, but wouldn't it be better as a comment since it doesn't really answer the question? If the current comments get deleted, people coming to this answer will have to figure out on their own that it's a joke, or ask an endless cycle of "explain this answer" questions in the comments. (to be clear I'm not against jokes at all, i love them, just not sure this type of answer is in the spirit of SE?) – bob Dec 5 '19 at 19:25
• This answer is the Platonic Ideal of MathOverflow answers. – Master Dec 6 '19 at 0:07

Index theorem of Atiyah and Singer closed a substantial field of research in the 1960s. I knew people who were working in this field, and had to switch the field of their research completely.

A more modern example is Louis de Branges proof of the Bieberbach conjecture. There was a large field of research, I would say a central field in analytic functions theory, which could be called "coefficients estimates". To be sure, it still exists, but nowadays it is considered marginal. Contrary to all expectations, the highly original proof of de Branges's theorem did not lead to a significant further development (so far).

Another commonly mentioned example is Hilbert's results in the theory of invariants. They closed the field in some sense, though not forever.

Darij Grinberg's description of this situation as "put to sleep" in his comment brings another similar example to my mind: in 1919/20 Pierre Fatou essentially "put to sleep" the wonderful field of holomorphic dynamics. He just did everything possible with the tools that existed at that time. The field was essentially sleeping until the early 1980s, when new, radically new tools were employed and some long standing problems were solved. (There is one isolated exception in this picture: Siegel's theorem of 1942, which also required a new tool, that is called KAM theory nowadays).

It also happens sometimes that a new breakthrough does not really close the field, but many people have to switch to another field because they are not equipped to understand the breakthrough. I do not want to give modern examples of such a sad situation, but according to Lev Pontryagin's own published recollections, he switched from topology to applied analysis in 1950s because the new abstract language introduced by the French revolutionized the area, and he could not stay in line with the modern development. (Pontryagin was one of the most prominent topologists of his time, and he was 42 years old in 1950.)

Another related phenomenon is an appearance of a definitive exposition of a subject which condemns much of the previous work to oblivion. An example is the book Orthogonal polynomials by Gabor Szego. It did not close the subject, far from it, but most people stopped reading and citing previous work. (Same thing that Euclid and Ptolemy did to their predecessors).

• The mirror symmetry attack on cluster algebras? I feel like the field of cluster algebras has moved out of the algebraic combinatorics mainstream after that paper came out, although I see other reasons for that to happen too. – darij grinberg Dec 3 '19 at 20:26
• My impression is that classical invariant theory has been put to sleep not by Hilbert's paper (which mainly proved qualitative results that weren't the priority of the British school) but by the unreadability and opaqueness of the British works. Hilbert's paper came out in 1890; the books of Weitzenböck and Turnbull in the 1920s. Would it really spend 30 years dying? My guess would be that it lost the competition for attention of young researchers against other parts of algebra that became much easier to learn due to better books (van der Waerden, later Bourbaki). – darij grinberg Dec 4 '19 at 9:15
• @Darij Grinberg: I agree: "put to sleep" is a better description. – Alexandre Eremenko Dec 4 '19 at 13:33

This is not, perhaps, a very large area, nor a complete "ending", but it was an interesting development in early semigroup theory that I think bears writing down.

Some background, first. A semigroup $$S$$ is a set with an associative binary operation $$\cdot : S \times S \to S$$. A semigroup is left cancellative if for all $$a, b, c \in S$$, we have $$ab = ac$$ implies $$b = c$$, and right cancellative if $$ba = ca$$ implies $$b = c$$. A semigroup is cancellative if it is left and right cancellative.

All groups are cancellative semigroups, but there are cancellative semigroups which are not groups (free semigroups, for example). Hence being cancellative is a necessary condition for a semigroup to embed in some group. A natural question is the following: does every cancellative semigroup embed in a group?

Anton Sushkevich initiated the study of cancellative semigroups in 1928. He was very interested in the problem of embedding cancellative semigroups in groups, and predicted that this very problem would become a central part of semigroup theory and produce a vast amount of new results in the area. This problem led to several publications by him and several others over the next few years, developing the theory of embedding cancellative semigroups in groups.

In [A. Sushkevich, "Про поширення півгрупи до цілої группы", Zapiski Khark. Mat. 4:12 (1935)], Sushkevich claimed a full affirmative answer -- being cancellative, he claimed, is sufficient for a semigroup to embed in a group!

But alas, in 1937, Malcev proved by way of example that there exists a cancellative semigroup which does not embed inside a group! In fact, he even provided a countable list of necessary and sufficient conditions for a cancellative semigroup to embed in a group, and showed that no finite sublist of this list is sufficient [A. Malcev, "On the Immersion of an Algebraic Ring into a Field," Math Annalen, Bd. 113, 5 Heft (1937)].

And yet, Sushkevich published a book in 1937, "The Theory of Generalised Groups" (of which very few physical copies still remain due to most copies being stored in Kharkov, Ukraine, during WW2, a city which was ruined in numerous battles during the war), which claimed to fix the errors in his original proof. The proof is tricky to read -- and while Sushkevich does acknowledge Malcev's example, he more or less only says "and so a sufficient condition for a semigroup to embed in a group is that it is cancellative and that it is not Malcev's example.

Malcev turned out to be right, and Sushkevich's proof was wrong (the operation for the "group" he claims to embed the semigroup in is not associative!). Indeed, Sushkevich spent the next few years attempting to remove any trace of his original publication, to the point where I would be surprised if any copy of the paper could be found. So a single counterexample was enough to -- if not "close" -- at least "deflate" the hype around cancellative semigroups.

Of course, as with any result, it only really served to spur further refinements and new lists of sufficient and necessary conditions, but its predicted central importance to semigroup theory fell short, and semigroup theorists mostly moved on to new pastures.

Bell's inequality killed the search for the elusive hidden variable theory that was supposed to complete Quantum Mechanics. IMO this qualifies because the result is mathematical.

• isn't it the experiment that measured the violation of Bell's inequality the thing that eliminated hidden variable theory? Bell's inequality by itself does not rule out that theory. – Carlo Beenakker Dec 3 '19 at 22:34
• @CarloBeenakker, Bell's inequality establishes that any hidden variable theory is necessarily incompatible with QM, and therefore eliminates the possibility that such theory would complete/supplement QM as QM stands. This killed the efforts to explain QM via hidden variables. The experiment, performed much later, only determines which of the now incompatible theories models the nature better. – Michael Dec 3 '19 at 22:54
• The linked article (under "Metaphysical aspects") includes a reference that non-local hidden variables are still considered viable in the form of De Broglie–Bohm theory. – Daniel R. Collins Dec 4 '19 at 4:03
• @Michael: Any local hidden variable theory. Nonlocal hidden variable models are trivially compatible with QM. – R.. GitHub STOP HELPING ICE Dec 4 '19 at 4:37
• Besides what @R.. says, there is also pilot-wave theory that is not excluded by QM and could be argued to be a kind of (non-local) hidden variable theory. – user21820 Dec 6 '19 at 3:22

Did Bertrand Russell not close down Frege's programme of logic with his observation of Russell's paradox? In this case, a simple observation essentially shut down Frege's programme of research, though he did carry on working on it and hoping that the ultimate aim of his project might still be possible in some sense.

• I think this answer can be expanded a bit (perhaps by someone more knowledgeable than myself). Russell and Whitehead continued work on the idea, but about the same time of the second edition of Principia Mathematica, Godel published his papers and that kind of killed the project, with some implications for Hilbert's open problems at the time. (I can't find a good source, but IIRC, Russell continued to work on his theory of classes, or at least revisited the idea a little some years later.) – Ben Burns Dec 6 '19 at 14:30
• You're right that there is definitely more to this story, perhaps someone else who knows more about it can go into more detail here. – Hollis Williams Dec 7 '19 at 17:15

I get the impression that Freedman's results from the 1980s "closed" the field of topological 4 manifolds, maybe in part because they were perceived as very difficult/intricate (see e.g. questions on MO like this one), but also for the reason 1 of the question asker (they "completely resolved" questions about these manifolds).

• There are still many basic open questions about topological 4-manifolds, especially ones with complicated fundamental groups (which are not covered by Freedman's work). It is true that progress has stalled on the subject to a certain degree. – Andy Putman Dec 4 '19 at 3:59

I don't know a particular paper to which to point, but, from a harmonic analyst's point of view, it seems to me that Weyl's results on compact Lie groups essentially closed the field of "representation theory of compact Lie groups." (I post this with great trepidation, because this or any answer seems an invitation to someone else to point out in what senses the field proposed is still alive and thriving—but that would be a fun thing to know, too!)

• I heard Elias Stein contributed something to this. – Bombyx mori Dec 4 '19 at 3:12

Alfred Tarski essentially 'closed' the ancient field of Euclidean geometry. Specifically, he provided a simple first-order axiomatisation of Euclidean geometry (where a model is a set of points endowed with a ternary 'betweenness' relation and a quaternary 'congruence' relation, obeying a particular set of axioms) and demonstrated that it is decidable: there exists an algorithm which can take any proposition expressed in the language of Tarski geometry and determine whether it is true or false.

• I don't agree with this answer at all. The theory of real-closed fields that underlies Tarski's theory of Euclidean geometry requires doubly-exponential time to decide, so numerous geometry problems remain unreachable by the decision procedure within human time scales. Furthermore, Tarski's axiomatization is not the only one; it is weaker than Hilbert's axiomatization, and the latter is in fact essentially incomplete. – user21820 Dec 6 '19 at 3:30
• Doubly-exponential in the number of times you alternate between universal and existential quantifiers. Are there any non-contrived geometry problems which alternate many times between universal and existential quantifiers? – Adam P. Goucher Jan 17 '20 at 18:21
• I don't know what exactly is the state-of-art now, but just a few years back I remember putting some elegant non-contrived geometry problem into some prover and it produced a corresponding polynomial with degree something like $48$, and did not finish proving the theorem (I don't recall what the problem was or how long I ran the prover for). And anyway, I can easily state some ancient geometry problems that you can't express in Tarski's axiomatization, such as Steiner's porism and Poncelot's porism. – user21820 Jan 19 '20 at 2:51

This is possibly a bit of a stretch (in relation to point 2); in that it concerns a modelling assumption which led to a variety of mathematical models which attempted to explain Marconi's transmission of radio waves across the Atlantic in 1901. The assumption was that the radio waves could propagate large distances beyond the horizon primarily via diffraction from the surface of the earth/ocean. The most well-known supporters of this idea (to mathoverflow users ... I assume), in the early twentieth century, were Poincaré and Sommerfeld, who amongst others, studied a variety of boundary value problems with this assumption playing a central role.

The paper that effectively closed this line of mathematical inquiry (almost 20 years old), was written by Watson (at the request of van der Pol) https://doi.org/10.1098/rspa.1918.0050. It is probably fair to say that experimental evidence also contributed to the demise of this idea (see Austin-Cohen reference in the paper ... and amateur experimental evidence) and an earlier paper by Nicholson which is referred to in Watson's paper.

The wider effects of this topic are discussed in detail Yeang's book (see review in the following link).