Modern results that are widely known, yet which at the time were ignored, not accepted or criticized What is your favorite example of a celebrated mathematical fact that had a hard time to become accepted by the community, but after overcoming some initial "resistance" quickly took on?
It can be a theorem, a proof method, an algorithm or a definition, that is

*

*widely known and

*very useful in the present day,

*less than 99 years old; this is in order to avoid examples from the very distant past, such the difficulties Grassmann's work had being accepted

but which at the time of its inception was not appreciated, misunderstood or ignored by the mathematical community, before it became mainstream or inspired other research which in turn became mainstream. This is a partial converse question to this one that asks for mathematical facts that were quickly accepted but then discarded by the community. This and this question are somewhat related, but former focuses on people (resp. their entire works, see Grassmann) not being accepted, rather then individual results, whereas the latter solely on famous articles rejected by journal; also, the results that are being mentioned in these links are often rather old and do not fit this question.

Example. Numerical optimization: The first quasi-Newton algorithm was discovered in 1959 and "was not accepted for publication; it
remained as a technical report for more than thirty years until it appeared in the ﬁrst issue of the SIAM Journal on Optimization in 1991" (Nocedal & Wright, Numerical Optimization).
But the algorithm inspired a slew of other variants, has been cited over 2000 times to this day and quasi-Newton type algorithms are still state-of-the-art in for certain optimization problems.
 A: Although more than 100 years old, this is my absolute favourite: Schlaefli's classification of regular polytopes in all dimensions using the Schlafli symbol,
see https://en.wikipedia.org/wiki/Schl%C3%A4fli_symbol.
It was probably not understood at his time, see https://en.wikipedia.org/wiki/Ludwig_Schl%C3%A4fli.
A: Sharkovsky's theorem on the coexistence of periodic cycles for continuous interval maps is a quite obvious example, I think, as the history section of the Scholarpedia entry explains.
A: I would guess that any great mathematical discovery made by physicists is at first met with great skepticism by mathematicians until being set on rigorous grounds and then considered with the uttermost respect. From the top of my mind, the Dirac delta function, the Verlinde formula, any mathematical concept with the word quantum, mirror or Feynman in it etc.
The following quote may hint at the problems encountered by mathematicians when it comes to asserting a truth coming from theoretical physics.
An absence of proof is a challenge; an absence of definition is deadly.

                                                          Deligne

The question asks for examples less than 99 years old but I can't resist mentioning the memoir of Fourier which competed without success for the prize of Academie des sciences, or simply the numerous controversies about infinitesimals at the dawn of modern analysis.
A: Heegner's published (1952) solution of Gauss class one problem (stating that there are only 9 imaginary quadratic number fields with class number=1) was not accepted until 1967. Only after Birch, Stark, and Baker (independently) found alternative solutions in 1967, Stark investigated Heegner's work and concluded that it was essentially correct.
A: The Selberg integral was proved in a 1944 paper of Selberg, after being stated without proof in a 1941 paper. The paper was in Norwegian, and was also in a journal that would have been of little interest to the research community:

This paper was published with some hesitation, and in Norwegian,
since I was rather doubtful that the results were new. The journal
is one which is read by mathematics-teachers in the gymnasium

This result was little-used, being used in one paper in 1953.
A closely related integral then appeared in random matrix theory. Mehta and Dyson gave a conjectural value for this integral, publicizing this conjecture as an open problem in a paper in 1963, a textbook in 1967, and the SIAM Review in 1974. However, no one remembered Selberg's work and thought to apply it.
Finally in 1976 Bombieri came across another similar integral when studying a different topic (prime numbers). He went to discuss his overall work on the distribution of prime numbers with Selberg, because of Selberg's expertise in number theory, and Selberg then mentioned his integral, which Bombieri used to solve his problem.
This was after Bombieri was informed by Spencer about the relationship of his integral to a third topic (the Coulomb gas), motivating him to ask Dyson about it, at which point Dyson explained the connection to random matrices, and thus Bombieri was able to prove the conjecture in random matrix theory as well.
Since then, the result has found further use and development, and is now widely-known.
My source for all these details is the paper The importance of the Selberg integral by Peter J. Forrester and S. Ole Warnaar
A: A controversial method was Appel & Haken's use of computers in their 1977 proof of the Four Color Theorem that had bested Kempe, Tait, and generations of graph theorists.  There's been a move to formal proof and, with improved computational power, memory, and techniques, results like Heule's determination in 2018 that 161 is the fifth Schur number.  The Computer-assisted proof Wikipedia page is pretty good, and there's a new StackExchange site on this topic on the way.
A: The classification theorem for three-dimensional convex polyhedra known as Steinitz's theorem first appeared in a 1922 publication of Ernst Steinitz. Because it did not use the language of graphs it remained obscure until it was given a graph-theoretic formulation in the 1960's.
A: Does acceptance of conjectures before they became theorems count?
Example 1. The Artin reciprocity law.  When Artin went around to other people describing what he was trying to show, nobody else believed it and they laughed at him for thinking it might be true. See here. This period of non-belief was only 3 years (the time it took Artin from formulation to proof).
Example 2. Modularity of elliptic curves over $\mathbf Q$. The original version by Taniyama in 1955 was expressed too broadly, but after that was fixed up it still took a bit of time for the idea to be generally accepted as plausible. For over 10 years, Shimura believed the conjecture but Weil, Serre, and others did not. See Lang's account of the history of the conjecture here.
Weil's identification of the conductor of an elliptic curve over $\mathbf Q$ with the level of the hypothetical associated modular form, in 1967, finally made the conjecture falsifiable and would explain some numerical observations if it were true, e.g., the smallest conductor of an elliptic curve  over $\mathbf Q$ is $11$ and the modularity conjecture would explain this because the modular curve $X_0(N)$ has genus $0$ for all $N < 11$, so no elliptic curve over $\mathbf Q$ could be the image of a morphism from $X_0(N)$ for $N < 11$.
A: The umbral calculus.  Even after Gian-Carlo Rota revived it, its significance was misunderstood by many.  No less a mathematician than Ira Gessel admitted this publicly, in his paper, Applications of the classical umbral calculus (arXiv version).

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.  The sequences that we consider here are defined by
exponential generating functions, and their most fundamental properties can be proved in
a straightforward way using these exponential generating functions. What is surprising
is that these sequences satisfy additional relations whose proofs require other methods.
The classical umbral calculus is a powerful but specialized tool that can be used to prove
these more esoteric formulas.

A: This one technically doesn't fit your stated criteria, but I think it's a good example in the same spirit.  Dan Shechtman's work on quasicrystals was initially strongly resisted, most famously by Linus Pauling, who snidely remarked, "There is no such thing as quasicrystals, only quasi-scientists." Earlier, similar discoveries by other scientists were similarly ignored or dismissed fairly quickly.
In this case, it seems that the mathematical work on aperiodic tilings, though well known and accepted in the mathematical community, was poorly understood or ignored or rejected as irrelevant by most scientists studying crystallography.
A: J. Howard Redfield anticipated many of the results in "Pólya enumeration" in his 1927 paper in the American Journal of Mathematics (https://doi.org/10.2307/2370675). But this work was largely forgotten until much later (the 1960s): see "The rediscovery of Redfield's papers" by Harary and Robinson (https://doi.org/10.1002/jgt.3190080202).
A: Grothendieck's inequality, now a fundamental result in functional analysis, with connections to computer science and quantum physics, had a difficult birth.
It was proved by Grothendieck in the paper Résumé de la théorie métrique des produits tensoriels topologiques, published in French in 1953 in an obscure Brazilian journal, only in very few copies, making it almost impossible to find.
The paper was almost completely ignored by the community, until it was rediscovered in 1968 by Lindenstrauss and Pelczynski who realized that in particular it contained answers to questions raised after its publication.
The story is explained in the first pages of the survey article by Gilles Pisier, Grothendieck's Theorem, past and present
A: My favorite example is Lu Jiaxi's work on large sets of disjoint Steiner triple systems and the generalization of Kirkman's schoolgirl problem. Although they might not fit well in the "widely known" criteria. But the story is fascinating anyway.

His paper on solving the generalized Kirkman's schoolgirl problem was ultimately rejected five years after he wrote them in 1961 and tried to publish them. In April 1979, in some journal issues of 1974 and 1975 that he managed to borrow from Beijing, he unexpectedly learned from a paper of Haim Hanani that the problem which he solved in his 1965 paper had been solved and first published in 1971 by Ray-Chaudhuri and R. M. Wilson, which was a big blow to him.


He went on to tackle the problem of large sets of disjoint Steiner triple systems. Zhu Lie, a professor of mathematics at Soochow University, realized the importance of his work and suggested that he submit it to the Journal of Combinatorial Theory, Series A. He wrote to its editorial board that he had essentially solved the problem, and the editors replied to him that if what he said was true, it would be a major achievement.

The Wikipedia article is a bit long and unpolished, with too many unnecessary details. But the overall read was remarkable.
A: Although strictly meanwhile more than 99 years old, a well-known example here
are the Julia- and Fatou sets. --
These sets were first investigated by Gaston Julia and Pierre Fatou in 1917/18,
but this work was more-or-less ignored until the invention of computers made it
possible to explore the beauties of these sets, and -- following the works of
Benoit Mandelbrot -- they became widely popularized in the 1980's.
