Fields of mathematics that were dormant for a long time until someone revitalized them

Question

I thought that the closed question here could be modified to a very interesting question (at least as far as big-list type questions go).

Can people name examples of fields of mathematics that were once very active, then fell dormant for a while (and maybe even were forgotten by most people!), and then were revived and became active again?

Here's an example to show what I mean. In the late 19th and early 20th century, hyperbolic geometry was an active and thriving field, attracting the attention of many of the best mathematicians of the era (for instance, Fricke, Klein, Dehn, etc). Fashions changed, however, and the subject was largely forgotten outside of textbooks. In the late '70's, however, Thurston introduced new ideas and showed that hyperbolic geometry was enormously important for the study of 3-manifolds, and now it and its offshoots have become central topics in low-dimensional topology and geometry.

Answer 1

The older theory of Hopf algebras, which grew out of algebraic topology as well as some purely algebraic theories, developed to the level of Sweedler's 1969 book and then became something of a backwater (at least as seen from the outside). But a generation later the study of quantum groups by Drinfeld, Jimbo, and their followers aroused interest in Hopf algebras on a far wider scale than the earlier algebraic work. The "dormancy" period here was not all that long, but I think it's fair to say that the earlier theory stayed mostly outside the mainstream (as measured by ICM programs, top journals, big grants, etc.).

Answer 2

Work of Julia, Fatou, Montel et al on complex dynamics that was largely forgotten or relegated to complex analysis textbooks until Douady – Hubbard and Mandelbrot revitalized it through the study of the Mandelbrot set supplemented by attractive computer graphics.

Answer 3

Knot theory. That seems like a canonical example: after a lot of interest up until the 1960s it became mathematical backwater in a way, but experienced an enormous surge in development with the discovery of the Jones polynomial and connections with physics (TQFT).

Related area: braid groups and mapping class groups. Besides connections with knots and physics, needs of low-dimensional topology and solution of some long-standing problems played a major role in the revival.

Answer 4

Symplectic and Contact geometries were invented in the 19th centuries as generalisations of the formalism classical mechanics and geometric optics, respectively. It seems to me both subjects soon died until the 1970-s, when Arnold became interested in the purely topological (as opposed to physics-related) aspects of these subjects and posed a few conjectures. Then, in 1980-s Gromov invented the method of J-holomorphic curves that allowed people to actually solve some problems in these subjects, and now both are very active. Nowadays, people have even invented ways to apply them to the study of general differentiable 3- and 4-manifolds.

Answer 5

Modular forms were actively studied by number theorists Hecke and Siegel in the 1930s, but it was not widely appreciated. Around the same time Hardy, in a series of lectures on Ramanujan's work delivered at Harvard in 1936, called modular forms -- as represented by Ramanujan's interest in the coefficients of the weight 12 form $\Delta(q) = \sum_{n \geq 1} \tau(n)q^n$ -- "one of the backwaters of mathematics". The study of modular forms basically died off in the 1940s and 1950s. It was revitalized by Weil, Shimura et al. in the 1960s. See the introduction to Lang's book on modular forms for some relevant historical remarks.

[EDIT: As Emerton points out in his comment below, the full quote by Hardy is actually more complimentary, so let me include it here: "We may seem to be straying into one of the backwaters of mathematics, but the genesis of $\tau(n)$ as a coefficient in so fundamental a function compels us to treat it with respect." This is at the start of Chapter X of Hardy's "Ramanjuan: Twelve Lectures on Subjects Suggested by his Life and Work."]

Answer 6

Back in 2008, I was trying to find a publisher to bring Dominic Welsh's classic book Matroid Theory back into print. I approached the American Mathematical Society, who solicited opinions from four experts. Here's what the experts said regarding the subject of matroid theory generally.

Matroid theory is not as active lately as it was a couple of decades ago, so I would not expect a large volume of sales.

The subject "matroid theory" is very narrow and not exactly active.

The field is not very active these days. Therefore a book about matroid theory won't find too many readers.

It depends on the philosophy of the series in which it should appear: classic YES, compelling topicality NO.

At the time, I personally didn't agree that matroid theory was inactive, but let's suppose that they were right and that matroid theory was dormant in 2008. There have been at least two major "revitalizing" developments since then. In 2013, Geelen, Gerards, and Whittle announced a proof of Rota's excluded minors conjecture. Around the same time, June Huh and his collaborators introduced the Hodge theory of matroids, solving a number of open problems (this latter development was cited by Richard Stanley in a recent interview as one of the four most influential advances in enumerative/algebraic combinatorics in the last 30 years). There is now an active matroid union blog from which you can get a sense of other recent developments in matroid theory. It would be hard to argue today that matroid theory is not active.

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UPDATE (2022): June Huh was awarded the Fields Medal, and the short citation focuses primarily on his work in matroid theory. So it is even harder now to argue that matroid theory is inactive.

Answer 7

The lambda calculus was first published in 1933 by Alonzo Church, intended to be an alternative to first-order logic. Two of his students, Kleene and Rosser, proved it inconsistent in 1935. In 1936, another student, Alan Turing, proved that a stripped-down version was equivalent in computational power to the Turing machine. It was pretty much "just another example of a Turing-equivalent language" for about 30 years.

In the late 1960s and 1970s, John McCarthy, Dana Scott, and Peter Landin (among others) revived the lambda calculus: John McCarthy by loosely basing LISP on it, Dana Scott by giving it a set-theoretic interpretation, and Peter Landin by developing a theoretical machine that - unlike Turing's - was similar to actual computers and a transformation into its machine instructions. These things together showed that the lambda calculus was not only good at encoding mathematical procedures as programs, but could be compiled into reasonable programs that run on actual machines. Guy Steele showed shortly after in his series of "Lambda the Ultimate" papers that the programs could be run efficiently, and that the lambda calculus easily models many familiar programming language constructs.

Today, the lambda calculus is THE theory of programming languages. Among the ways to describe algorithms, its mathematical purity is unrivaled.

Compared to other mathematical areas, 30 years isn't a long time for something to lay dormant. But consider that Computer Science has only been around for about 70!

Answer 8

Modal Logic.

This goes back to Aristotle. It was picked up by Medieval and Arab philosophers (often associated with proofs of the existence of God) but I don't think it was taken very seriously by mathematicians until the 20th century. Kripke provided nice semantics for modal logics in terms of possible worlds in the 50s (I think) and since then the subject has blossomed. Nowadays modal logics are a commonplace tool in computer science.

Answer 9

This might not be exactly what you're asking for but I think it's close: Manjul Bhargava's generalizations of Gauss's composition law to higher composition laws. While Gauss's composition law did not exactly languish in obscurity, it is clear that Bhargava's stunning work has revitalized a classical subject. Perhaps one could argue that this is really a case of a long-standing open problem finally being solved, but my sense is that it's more accurate to say that Bhargava found unsuspected treasures in deceptively familiar territory.

Answer 10

As Jim Humphreys has suggested in the comments, practically all of Gian-Carlo Rota's career could be described as breathing new life into unjustly neglected subjects: Möbius functions of posets, invariant theory, lattice theory, etc. For the purposes of MO, let me single out the umbral calculus as a specific subject that languished and was revived by Rota. For anyone who is skeptical of the power of umbral calculus, I recommend Gessel's paper on applications of the classical umbral calculus. Gessel writes:

When I first encountered umbral notation it seemed to me that this was all there was to it; it was simply a notation for dealing with exponential generating functions, or to put it bluntly, it was a method for avoiding the use of exponential generating functions when they really ought to be used. The point of this paper is that my first impression was wrong: none of the results proved here (with the exception of Theorem 7.1, and perhaps a few other results in section 7) can be easily proved by straightforward manipulation of exponential generating functions.

Answer 11

Branko Grünbaum wrote in 1978 lecture notes called "Lectures on Lost Mathematics" Grünbaum talks about areas of geometry that went "underground". The topics discussed there and Grünbaum philosophical comments (e.g. p. 15 of the pdf file where the original manuscript begins) are quite relevant to the topic of the question.

Some topics discussed by Grünbaum were "revived" in some cases because of these lecture notes and in other cases independently.

Answer 12

The study of generalized symmetries of differential equations was initiated by Emmy Noether in the context of her famous theorem but by and large the field lay dormant until it was revitalized by the discovery of the equations integrable via the inverse scattering transform (i.e., roughly until 1970s).

Answer 13

This wikipedia link on dessin d'enfants says the following.

Early proto-forms of dessins d'enfants appeared as early as 1856 in the Icosian Calculus of William Rowan Hamilton in modern terms, Hamiltonian paths on the icosahedral graph. Recognizable modern dessins d'enfants (and Belyi functions) were used by Felix Klein, which he called Linienzüge (German, plural of Linienzug “line-track”, also used as a term for polygon. Dessins d'enfant in their modern form were then rediscovered over a century later and named by Alexander Grothendieck in 1984 in his Esquisse d'un Programme.

Answer 14

This example may not be that of a whole field but I think it illustrates an important result that lay dormant for a very long time. A natural question in the theory of graphs is when is a graph the vertex-edge graph of a 3-dimensional convex polyhedron? It turns out that this question was in essence answered by Ernst Steinitz in 1922. However, Steinitz did not use a graph theory framework for his work. As a consequence, almost no one noticed what he had accomplished. Almost no references to Steinitz's work was made until 1962 and 1963 when Branko Grünbaum and Theodore Motzkin wrote two papers where they mentioned what Steinitz had done but reformulated it using graph theory terminology. The result in these terms, now known as Steinitz's Theorem states that a graph is the vertex-edge graph of a convex 3-dimensional polyhedron if and only if the graph is planar and 3-connected. A good place to read about this is in Grünbaum's book: Convex Polytopes (2nd edition). Grünbaum (and others) went on to produce many papers that exploited Steinitz's Theorem in many directions. One way to think of what was accomplished here was that to study the combinatorial properties of 3-dimensional convex polyhedra one does not have to think in 3-dimensions but only in 2 dimensions.

Answer 15

Analytic number theory has gone through several cycles of high and low activity. Around 1900 complex analysis solved lots of hitherto unsolvable problems. In the 20's there was much less activity, but in the 30's and 40's there was odd Goldbach, Vinogradov's bound for exponential sums, Linnik's theorem, Siegel's zero. Then again much less activity till the large sieve and density results around 1970. After the large sieve there was a decline, and in 1990 so many mathematicians believed that analytic number theory was dead that most positions were converted to more "modern" subjects, in particular arithmetic geometry. And then suddenly prime detecting sieves, subconvexity, additive combinatorics, explicit congruencing, and we see that the whole theory is well alive.

Answer 16

Some historians have speculated that classical Greek geometers used "hidden" analytic methods to discover results, which they then reconstructed synthetically. Further, it seems that Archimedes was further along towards calculus (though I gather this might be a bit exaggerated) than had been thought. In any case, the gradual decline of Aristotelean finitism and dismissal of empiricist epistemological constraints on geometrical reasoning in the 17th/18th century and the subsequent period of mathematical advance might be an instance of revitalization/rediscovery rather than of completely new developments, though these are claims in need of more careful historical argument than I'm in a position to give.

Answer 17

I've heard that finite group theory was pretty big in the late 19th century, lay dormant for a while, and had a big increase in activity around 1960 when cases of the Odd Order Theorem started falling. After the theorem was proved, a full classification of finite simple groups looked (to some experts) like a more reasonable goal than before.

(I am somewhat unqualified to elaborate - edits are welcome.)

Answer 18

Hyperdeterminants which are generalisations of determinants to multidimensional hypermatrices were first found and developed by Cayley in the mid 19th century and were actively studied until around 1900. Then general results on invariants such Hilbert's basis theorem made them look redundant. For most of the 20th century few mathematicians would even have recognised Cayley's simplest hyperdeterminants if they came upon them.

Then there were a series of rediscoveries of these objects in areas of mathematics and physics, (e.g. hypergeometric functions, Diophantine equations, qubit entanglement and string theory) Now people are starting to look at them again and realize that they are useful and not yet fully understood.

Answer 19

q-special functions (basic hypergeometric functions) were developed between the end of the XIXth and beginning of the XXth century. Then remained somewhat in the background as a very peculiar math gadget. When it turned out that they play a relevant role in the representation theory of quantum groups, towards the end of the 80's, they came back into play and enjoyed a very lively period.

Answer 20

This one is debatable but it comes with a wonderful quote. In ¨Geometry and the imagination¨ Hilbert and Cohn Vossen wrote, It might be mentioned here, that there was a time in which the study of configurations (incidence configurations between points and lines) was considered the most important part of all geometry. Today we have the Kakeya conjecture, the theorems of Szemeredi-Trotter, of Sylvester-Gallai and a long list of relatives in discrete geometry, additive combinatorics and harmonic analysis.

Answer 21

C.S. Peirce was lecturing on what he called the “laws of information” as early as 1865–1866 and later gave a simple form of logarithmic measure for the information content of a logical constraint. Of course, he gave these lectures at those colonial backwaters known as Harvard College and the Lowell Institute, so it's no surprise these seeds of information theory took so long to sprout.

Answer 22

Polynomial Chaos was developed in the late 30s by N. Wiener, but went more or less unnoticed until Ghanem & Spanos picked up on it for use in finite element analysis in the 80s and 90s. In some ways it still may be an under-utilized approach, given the dominance of the Itô and Stratonovich calculi.

Answer 23

De Morgan established a calculus of binary relations in 1860. Charles Peirce turned out to the subject in 1870, and found most of the interesting equational laws of relation algebra. The subject fell into neglect between 1900 and 1940, to be revived by Alfred Tarski.

Answer 24

In Pi and the AGM by Borwein and Borwein we find the following remark of Klein [1928]:

When I was a student, abelian functions were ... the uncontested summit of mathematics ... And now? The younger generation hardly knows abelian functions.

How did this happen? In mathematics, as in other sciences, the same processes can be observed again and again. First, new questions arise and draw researchers away from the old questions. And the old questions, just because they have been worked on so much, need ever more comprehensive study for their mastery ...

And so there is nothing for it but to collect together the old subjects in good references ... so that later developments may continue them, if fate should so decree.

Answer 25

Kervaire invariant theory (that is: the question of when a given framed manifold can be converted into a sphere via surgery) was dormant from 1969 until 2009, when Hill, Hopkins, and Ravenel announced a proof (published in 2016) of the Kervaire Invariant Theorem. They now have a wonderful book about this work, where I learned the quotations below.

After a flurry of activity in the 1950s and 60s, the problem was solved in all dimensions other than those of the form $2^{j+1}-2$. The Kervaire invariant of a manifold is always 0 or 1, and if it's 0 then there's no obstruction to framed surgery, so $M$ is framed cobordant to a homotopy sphere.

In dimensions of the form $2^{j+1}-2$, Browder (1969) proved that there exists a framed manifold of dimension $2^{j+1}-2$ with nontrivial Kervaire invariant if and only if the element $h_j^2$ in the Adams spectral sequence (at the prime 2) is a permanent cycle, i.e., survives to an element $\theta_j$ in $\pi_{2^{j+1}-2}^S$. It was known that such manifolds do exist for $j < 6$, i.e., in dimensions 2, 6, 14, 30, and 62. But there were infinitely many dimensions (126, 254, ...) where the answer wasn't known.

Just before the Hill, Hopkins, Ravenel announcement, Victor Snaith published a book with the following quotes:

As ideas for progress on a particular mathematics problem atrophy it can disappear. Accordingly I wrote this book to stem the tide of oblivion.

He also wrote:

In the light of the above conjecture and the failure over fifty years to construct framed manifolds of Arf–Kervaire invariant one this might turn out to be a book about things which do not exist. This [is] why the quotations which preface each chapter contain a preponderance of utterances from the pen of Lewis Carroll.

In 2009, Hill, Hopkins, and Ravenel proved that, indeed, there are no manifolds of Kervaire invariant one in those dimensions (for $j \geq 7$, leaving the case $j=6$, corresponding to dimension 126, open). The field is no longer dormant. People are working hard to compute stable homotopy groups (especially, using new techniques from motivic homotopy) to answer this question in dimension 126. A 2017 Annals paper by Wang and Xu built on this story to prove there are no exotic smooth structures on spheres in dimensions 5, 6, 12, 56, and 61.

Answer 26

I believe that at the time that Stokes began studying fluid mechanics it was something of a dead subject and his contributions which we all know played a large part in reviving interest in it.