Examples of simultaneous independent breakthroughs I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. Two examples come to mind:


*

*Prime Number Theorem. This was proved by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896.

*Sum of three fourth powers equals a fourth power. In 1986, Noam Elkies proved that there are infinitely many integer solutions to $a^4 + b^4 + c^4 = d^4$. His smallest example was $2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$. Don Zagier reported that he found a solution independently just weeks later.


Can you give other instances?
 A: Van der Waerden's conjecture on the permanent of a doubly stochastic matrix had been open for 50 years when it was proved independently by Falikman and Egorychev.
A: The independence of the Parallel Postulate had been open for about 2000 years when its independence had been independently by Bolyai, Lobachevski and Gauss around 1830; Gauss never published his proof, but only claimed to have solved the problem when he was informed about Bolyai's proof.
A: The theory of NP-completeness was simultaneously and independently developed in North America and the U.S.S.R.
A: Recently I reviewed
Shmerkin, Pablo On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions. Ann. of Math. (2) 189 (2019), 2, 319--391
and obverved that another proof of Furstenberg's intersection conjecture is given in 
Wu, Meng, A proof of Furstenberg’s conjecture on the intersections of ×p and ×q-invariant sets, Ann. of Math. (2) 189, 3, 707-751, (2019)
The methods of the proofs seems to be quite different.
A: The "class number one" problem for imaginary quadratic fields was posed by Gauss in 1798, settled independently by Baker and Stark in 1966/7. Gauss had found nine imaginary quadratic fields with class number one (which, in this context, is equivalent to unique factorization into primes) and conjectured there were no others; Baker and Stark proved this. 
(There was also a proof by Heegner some years earlier than the work of Baker and Stark, but this proof had some minor gaps and was not generally accepted until Stark showed how to fill the gaps in 1969.)
A: The equivalence of nondeterministic logarithmic complexity class (NL) to the class whose complements are in NL (co-NL) was proven independently by Neil Immerman (University of Massachusetts, Amherst, MA, USA) and Róbert Szelepcsényi (Comenius University, Bratislava, Slovakia).  This result is known as the Immerman–Szelepcsényi theorem and they shared the 1995 Gödel Prize for it.
A: One of the most entertaining seminar talks I ever attended was by Michael Atiyah on the moment map on a Lie group. As the talk progressed, the people in the front row became more and more agitated, until Raoul Bott finally spoke up, said something like, "Michael, these guys just proved the same theorem last week!", and pointed to Victor Guillemin and Shlomo Sternberg.
A: During the summer of 2001, Crew's conjecture in the theory of $p$-adic differential equations was simultaneously proved in three very different ways by André, Kedlaya, and Mebkhout.
A: The word problem for groups was shown to be undecidable by Pyortr Novikov in 1955 and William Boone in 1958. Since there was practically no academic communication between their home countries (USSR and USA, respectively) at the time, the consensus is that the discoveries were independent.
A: In 1992, Feder and Vardi conjectured that for every constraint language $\Gamma$, the constraint satisfaction problem with constraints from $\Gamma$ is either solvable in polynomial time or is $\mathrm{NP}$-complete.  Progress was made on special cases and related problems but not a whole lot happened until Bulatov and Zhuk independently announced proofs of the conjecture in March and April 2017.
As I recall, a third proof was announced at around the same time, but was subsequently found to be erroneous and withdrawn.
A: The HOMFLY or HOMFLYPT polynomial is a clear example. The polynomial was introduced in "A new polynomial invariant of knots and links", and on the first page there is a footnote:

Editor's Note. The editors received, virtually within a period of a few days in late September and early October 1984, four research announcements, each describing the same result—the existence and properties of a new polynomial invariant for knots and links. There was variation in the approaches taken by the four groups and variation in corollaries and elaboration. These were: A new invariant for knots and links by Peter Freyd and David Yetter; A polynomial invariant of knots and links by Jim Hoste; Topological invariants of knots and links, by W. B. R. Lickorish and Kenneth C. Millett, and A polynomial invariant for knots: A combinatorial and an algebraic approach, by A. Ocneanu.
It was evident from the circumstances that the four groups arrived at their results completely independently of each other, although all were inspired by the work of Jones (cf. [10], and also [8, 9]). The degree of simultaneity was such that, by common consent, it was unproductive to try to assess priority. Indeed it would seem that there is enough credit for all to share in.
Each of these papers was refereed, and we would have happily published any one of them, had it been the only one under consideration. Because the alternatives of publication of all four or of none were both unsatisfying, all have agreed to the compromise embodied here of a paper carrying all six names as coauthors, consisting of an introductory section describing the basics written by a disinterested party, and followed by four sections, one written by each of the four groups, briefly describing the highlights of their own approach and elaboration.

The result also appears in work by Józef H. Przytycki and Paweł Traczyk, hence the PT.
The reason for simultaneous discovery is quite clear: the HOMFLYPT polynomial is a relatively straightforward generalization of the (at the time) recently introduced Jones polynomial.
A: Obviously, the most famous example would be Newton-Leibniz development of Calculus, but there is also the case of Lagrange and Euler when developing the calculus of variations. A list can be found here. 
A: The Gelfond-Schneider Theorem, if $a$ and $b$ are algebraic numbers with $a\ne0,1$, and $b$ irrational, then any value of $a^b$ is a transcendental number, was proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
A: One of the most famous ones seems to be Newton vs. Leibniz, see e.g. https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy https://www.springer.com/de/book/9783319725611 
A: The solution of Hilbert's  nineteenth problem, in 1957, by Ennio De Giorgi and John Nash, few months later.
A: The angel problem was posed in 1982, and little progress was made until it was solved independently and almost simultaneously in 2006 by four different people. To borrow from the Wikipedia article, the question is to determine the winner of a certain game: 

The game is played by two players called the angel and the devil. It is played on an infinite chessboard. The angel has a power $k \in \mathbb{N}$ specified before the game starts. On each turn, the angel flies to a different square whose distance from its current square is at most $k$ in the infinity norm. The devil, on its turn, may permanently block any single square not containing the angel. The angel may fly over blocked squares, but cannot land on them. The devil wins if the angel is unable to move. The angel wins by surviving indefinitely.

The problem was first published in the book Winning Ways for Your Mathematical Plays by Berlekamp, Conway, and Guy and therefore became fairly well known among people interested in recreational mathematics. However, arguably the reason for the sudden appearance of four independent solutions in 2006 was the publication of Peter Winkler's book Mathematical Puzzles: A Connoisseur's Collection, which listed the angel problem as an unsolved problem, and presumably sparked the interest of a lot of people who had either not heard the problem before, or at least had not given it much thought.
A: The Entscheidungsproblem was a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms.
The problem was solved independently by Alonzo Church and A. M. Turing in 1936. Church's solution was based on lambda calculus, whereas Turing's approach was based on hypothetical computational devices now known as Turing Machines.
On a related note, a definition of computation was also provided independently by Emil L. Post in 1936, and is equivalent to the the other two approaches.
A: The near simultaneous solution to Serre's conjecture on projective modules over polynomial rings by Quillen and Suslin independently comes to mind.
A: Recent example in analytic number theory:
http://annals.math.princeton.edu/2016/183-3/p03
and
http://annals.math.princeton.edu/2016/183-3/p04
A: A very recent example appears to be the simultaneous discovery of functorial (and "easy") resolution of singularities in characteristic 0 by McQuillan-Marzo (arXiv:1906.06745) respectively Abramovich-Temkin-Włodarczyk (arXiv:1906.07106).
A: I'm not sure if it satisfies the criteria of "a long time with little progress", but one example of a simultaneous mathematical discovery is the discovery of the hook length formula. Here is how Bruce Sagan describes it in his book The Symmetric Group:

One Thursday in May of 1953, Robinson was visiting Frame at Michigan State University. Discussing the work of Staal (a student of Robinson), Frame was led to conjecture the hook formula. At first Robinson could not believe that such a simple formula existed, but after trying some examples he became convinced, and together they proved the identity. On Saturday they went to the University of Michigan, where Frame presented their new result after a lecture by Robinson. This surprised Thrall, who was in the audience, because he had just proved the same result on the same day!

A: Abel and Galois were working independently and did not know each other. Their approaches were different but they solved (among other things) an old and famous problem.
There are many other examples: Abel and Jacobi (elliptic functions),
Bolyai, Lobachevski and Gauss independently developed non-Euclidean geometry.
A: Richard Friedberg and Albert Muchnik independently and almost simultaneously solved Post's problem.
A: A conjecture from the 1950s (arising from a result of Kaplansky) was that for any compact Hausdorff space $X$,  every algebra homomorphism from $C(X)$ into another Banach algebra is automatically continuous. This conjecture was refuted in the late 1970s, independently, by H. Garth Dales and Jean Esterle: a very short announcement which outlines some of the ideas can be found in
H. G. Dales and J. Esterle, Discontinuous homomorphisms from C(X).
Bull. Amer. Math. Soc. 83 (1977), 257–259
Both approaches assumed the continuum hypothesis; later work of Solovay exhibited models of ZFC in which the original conjecture has a positive answer.
A: Ramanujan graphs were independently constructed by Margulis and by Lubotzky–Phillips–Sarnak.
A: The theoretical basis of Linear Optimization has been independently developed in the time of the second world war by L.V. Kantorovich and by G.B. Dantzig.
A: Hillel Furstenberg and Robert Zimmer simultaneously proved a powerful structure theorem for measure preserving dynamical systems: exposition at Tao’s blog.  Furstenberg’s article also contains a proof of a conjecture of László Lovász, on the density of a set of integers, no two of which differ by a perfect square.  This was proved simultaneously and independently with a different method, by András Sárközy.
A: Ray and Singer conjectured that their analytic torsion should carry the same information as Reidemeister torsion. This was then proved independently by Cheeger and Müller, see wikipedia.
But maybe this does not count because the phase of inactivity was not that long ...
A: A completely new algorithm for calculating $\pi$ was independently discovered by Salamin and Brent in 1976. This iterative algorithm, which doubles the number of correct digits with each iteration, is based on work of Gauss (arithmetic-geometric means) that had been neglected for 170 years. 
A: Last month (September 2020) Chodosh and Li uploaded a preprint showing that a closed aspherical manifold of dimension $4$ or $5$ does not admit a Riemannian metric with positive scalar curvature.
On exactly the same day Gromov uploaded a preprint which used similar ideas to generalize the result of Chodosh and Li to higher dimension (again, as usual, the fact that they were both submitted on the same day was almost certainly coordinated between the authors and not a coincidence).
