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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.

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Surely somebody here will want to comment on the periodic rebirth of invariant theory? –  Jim Humphreys May 11 '10 at 17:44
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Rota has interesting things to say about this phenomenon, but I can't quite remember where he wrote them down... –  Qiaochu Yuan May 11 '10 at 20:15
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@Qiaochu: Yes, I was thinking of Gian-Carlo Rota when I wrote that line. He actually wrote things down in a lot of places, having passionate views about invariant theory and its neglect. My own view of the subject has always been more tentative, but his personality is unforgettable for those of us who encountered him. –  Jim Humphreys May 11 '10 at 22:24
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Actually, another example just came to me. Dehn proved that the mapping class group of a surface was generated by a finite number of so-called Dehn twists. This was forgotten for many years, and in the '70's Lickorish rediscovered the result (for a while, these mapping classes were known as "Lickorish Twists" <grin>). This was the beginning of an enormous amount of work. However, if you go back to Dehn's papers, you will find many of the things people spent the next 10 or so years rediscovering (for instance, the famous "lantern relation" between mapping classes). –  Andy Putman May 12 '10 at 2:49
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+1. I hope that people confronted with potentially interesting but ultimately flawed questions will do this in the future rather than trying to keep the original question open. –  Harry Gindi May 12 '10 at 8:05

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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.

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It can be said that the work of Julia, Fatou, Lattes etc. revived earlier work by Schroeder, Koenigs, Boettcher etc. on iteration theory. A good account is Daniel Alexander's book on history of complex dynamics and its follow-up, Early Days in Complex Dynamics: A history of complex dynamics in one variable during 1906-1942 Daniel S. Alexander, Drake University, Des Moines, IA, Felice Iavernaro, Università di Bari, Italy, and Alessandro Rosa A co-publication of the AMS and the London Mathematical Society –  Margaret Friedland Nov 3 '11 at 18:27

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.).

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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."]

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I think String Theory also kicked new life into modular forms encouraging physicists to revisit a bunch of classical work, reading new physical meaning into it. For example, $\Delta$ plays (part of) the role of a generating function for the number of physical states corresponding to each energy level. –  Dan Piponi May 11 '10 at 17:55
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One should be careful in interpreting Hardy's remark, which is often quoted in the spirit that it is here. I think a careful reading of the text will show that Hardy says that "it may seem" that one is entering the backwaters of mathematics, but his intention (in the rest of that chapter, which is the last in his book on Ramanujan) is to show that in fact it is not the backwaters at all. –  Emerton May 11 '10 at 17:57
    
One should also consider the representation theoretic point of view on more general automorphic forms which began to be developed by Gelfand and others in the 50s and 60s, as well as Selberg's work, which together formed the base of Langlands massive work on Eisenstein series. I guess like any deep mathematical subject, the detailed history of its development is complicated. But I don't think one can deny that the connection between modular forms and arithmetic was dormant for quite some time. –  Emerton May 11 '10 at 18:01
    
Matt: Hardy's chapter on Ramanujan's work on tau(n) is not the last one in his book: it is X out of XII. –  KConrad May 11 '10 at 18:29
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It dismays me to register my age in this way, but... Around 1970, "automorphic forms and L-functions" was not the booming business it is now. Langlands' results were privately circulated [criticism possible here], Shimura's results about generation of classfields and Hasse-Weil zeta functions were ... difficult, Weil's extension of Hecke's converse theorem was a bit baroque, Gelfand-Fomin-PiatetskiShapiro's understanding of the power of repn theory in automorphic forms was new, ... as was Harish-Chandra's repn theory... Godement's repn theory... Ack!!! It confused me then, as now... :) 8 chrs –  paul garrett Jun 22 '11 at 22:59

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.

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Maybe this should go back to Lord Kelvin and Tait –  Ian Agol Mar 9 '11 at 22:33
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o.O < ponders the backwaters of 1984--1960 knot theory inhabited by D. Gabai, W. Thurston, D. Rolfsen, H. Hilden, J. Montesinos, J. Birman, C. Gordon, L. Kaufmann, R. Riley, W. Whitten, L. Neuworth, K. Murasugi, R. Fox.......... > –  Lee Mosher Apr 1 '12 at 20:59

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.

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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.

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again related to invariant theory and Cayley's hyperdeterminants... –  Abdelmalek Abdesselam May 11 '12 at 20:46

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.

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An interesting point and I'm gonna bring it up to Dr.Kripke in his seminar next semester at the CUNY Graduate Center. –  Andrew L May 12 '10 at 0:44
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Leibniz wrote a lot on what (today) is recognized as modal logic. Russell greatly misunderstood some of Leibniz's work because 'modal logic' had not yet been properly invented. –  Jacques Carette May 14 '10 at 18:20
    
@Jacques In fact, I take Russell's comments on the subject as evidence the subject was dormant. –  Dan Piponi May 14 '10 at 23:20
    
Considering the little Aristotle I have read, I find it hard to believe he had anything worthwhile to say about modal logic. It is not sufficient to state that some things are "necessary" and others are "possible", many people have done that, but one must also define at least a rudimentary set of formal rules for manipulating negations and conjunctions of these statements. –  Ron Maimon Aug 1 '11 at 13:27
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Aristotle did in fact give some rules for manipulation of propositions with modalities. For example, he gave the rule that "necessarily P" is equivalent to "not possibly not P" as well as a bunch of entailment rules. plato.stanford.edu/entries/aristotle-logic/#Syl –  Dan Piponi Aug 1 '11 at 17:16

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).

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The same theory of integrable systems also picked up on the much earlier work of Jacobi, Neumann and Sofia Kovalevskaya on explicit integration of differential equations by theta functions - see, for example, Mumford's "Tata lectures on Theta". –  Victor Protsak May 12 '10 at 2:19

Branko Grunbaum wrote in 1978 lecture notes called "Lectures on Lost Mathematics" Grunbaum talks about areas of geometry that went "underground". The topics discussed there and Grunbaum 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 Grunbaum were "revived" in some cases because of these lecture notes and in other cases independently.

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Hi Gil, thanks for this great link. While many conjectures in the notes are resolved by now, I am afraid some of this material is again "overly neglected" (in Grunbaum's words). That's really too bad... –  Igor Pak May 12 '10 at 9:21
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Grünbaum has in two other publications shown the value of taking a "fresh look" at geometrical/combinatorial problems that had been considered by geometers/combinatorialists in the past (19th century), using the power of more recent ideas and methods. These are: Arrangements and Spreads (American Mathematical Society, 1972) and the very recent book: Configurations of Points and Lines (AMS, 2009). Arrangements and Spreads has stimulated many new results in geometry and computational geometry. –  Joseph Malkevitch May 12 '10 at 14:03

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.

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The mystery deepens when one considers the discovery of the power series for sine, cosine and inverse tangent (with the corollary that $\pi/4=1-1/3+1/5-1/7+\cdots$) by Madhava and others in India around 1400. –  John Stillwell May 11 '10 at 23:16
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John: Do you have a reference for this? –  Łukasz Grabowski Oct 29 '10 at 13:06
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Exposition in Mathematics Magazine 63 (1990) 291--306 "The Discovery of the Series Formula for pi by Leibniz, Gregory and Nilakantha" by Ranjan Roy –  Gerald Edgar Mar 9 '11 at 14:49

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.

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Umbral calculus and invariant theory are quite related since the former is one way of making sense of the symbolic method used in the latter. One could surmise that this is why G-C Rota was interested in both subjects at the same time. –  Abdelmalek Abdesselam Mar 9 '11 at 16:21

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.)

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This is probably overstated. The work of Frobenius, Schur, Burnside, Brauer, and others in the first decades of the 20th century took the subject in new directions. This grew out of 19th century invariant theory but opened the door (by the time of the Brauer-Fowler paper in 1955 on centralizers of involutions) to Feit-Thompson and the acceleration of the classification project. Representation theory of finite groups has kept developing in many directions, alongside the structure theory which levelled off more after 1980. –  Jim Humphreys May 11 '10 at 17:30
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You should also remember Philip Hall! –  Yiftach Barnea May 11 '10 at 19:36
    
Yes, Hall was a major influence, especially in British group theory. I should emphasize that the first half of the 20th century, with its world wars, depression, and other upheavals, was not a flourishing period for mathematics. There were relatively few people doing real research, no "institutes" before IAS, disbanding of European research centers, few journals, slow communication. Even so, finite group theory, combinatorial group theory, Lie theory all made important progress. But it's true that finite group theory became less visible than other subjects in that period. –  Jim Humphreys May 11 '10 at 21:02
    
Actually, I think that in some sense, (finite) group theory went out of fashion after the announcement of the classification, especially in America. I would say that only now in the last decade or two it is coming back. Is this just my feeling? –  Yiftach Barnea May 11 '10 at 22:07
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The structure of finite groups has always been a rather specialized field ignored by most mathematicians. But the ordinary and modular representation theory has continued to have a fairly high profile, though here too it's somewhat of an acquired taste and usually not directly applicable elsewhere. George Lusztig has for example reshaped the agenda for groups of Lie type and attracted new interest in the area. Here the modular theory is still far from being understood. The classification of simple groups is another matter, having reached highest intensity around 1980. –  Jim Humphreys May 11 '10 at 22:36

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!

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If I remember right, LISP was invented in the late 50's. Something like 1958 or so. –  Asaf Karagila Nov 3 '11 at 17:48
    
Oh, you're right. 1958 it is. My point still stands, though: programming languages, as a mathematical field, was pretty much dead until Scott, Landin and Steele revived the lambda calculus and connected the mathematics to the machines. That makes John McCarthy a forerunner, a voice crying in the wilderness... "LAMBDA!" –  Neil Toronto Nov 6 '11 at 23:27

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.

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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.

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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.

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Indeed there was renewed interest in these object, especially after the book by Gelfand, Kapranov and Zelevinsky. There was also a series a massive papers by Jouanolou. –  Abdelmalek Abdesselam Mar 9 '11 at 16:16

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.

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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.

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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 Itō and Stratonovich calculi.

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Did you mean Wiener? –  mathphysicist May 11 '10 at 22:57
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@mathphysicist: ain't community wiki great? :) –  Willie Wong May 12 '10 at 0:53
    
My typo rate for mathematicians' names is 2/2 the last couple days... :( –  Ed Gorcenski May 12 '10 at 2:05

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.

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Point set topological methods are seeing a revivial in a very unexpected place:number theory. The use of ultrafilter methods as well as "tailored" topologies on the integers to produce topological proofs of number theoretic results-pioneered by Kevin Broughan and being pursued by Melvyn Nathanson,myself and several others-are causing a renewed interest in point set methods among number theorists lately.

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Can you give an example of a new number theory result proved in this way? –  BCnrd May 12 '10 at 1:33
    
The key bad word in this answer is methods. The question asks for fields of mathematics, probably because examples of methods that waxed and waned are just too easy to find; it must be commonplace in every field. –  Thierry Zell Jan 22 '11 at 1:40

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