Examples of simultaneous independent breakthroughsExamples of great mathematical writingFundamental...
Examples of simultaneous independent breakthroughs
Examples of great mathematical writingFundamental ExamplesWhat are examples of theorems get extensions based on simple observation?Has mathoverflow yet led to mathematical breakthroughs?What are some examples of narrowly missed discoveries in the history of mathematics? Examples of “folk theorems”What are examples of theorems which were once “valid”, then became “invalid” as standard definitions shifted?Examples where adding complexity made a problem simplerFamous examples of PhD advisors younger than their studentWhat are the current breakthroughs of Geometric Complexity Theory?
$begingroup$
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?
soft-question ho.history-overview big-list
$endgroup$
add a comment |
$begingroup$
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?
soft-question ho.history-overview big-list
$endgroup$
2
$begingroup$
This happens frequently. Search for the word "independently" in abstracts or in the conclusion (or introduction) of papers. My name appears in Algebra Universalis because Libor Polak acknowledged my near simultaneous solution to a problem in logic he solved. (In this case though, the problem had not been around long.) Gerhard "Or Look Up 'On Hyperassociativity'" Paseman, 2019.07.26.
$endgroup$
– Gerhard Paseman
10 hours ago
add a comment |
$begingroup$
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?
soft-question ho.history-overview big-list
$endgroup$
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?
soft-question ho.history-overview big-list
soft-question ho.history-overview big-list
edited 5 hours ago
community wiki
4 revs, 3 users 65%
Marc Chamberland
2
$begingroup$
This happens frequently. Search for the word "independently" in abstracts or in the conclusion (or introduction) of papers. My name appears in Algebra Universalis because Libor Polak acknowledged my near simultaneous solution to a problem in logic he solved. (In this case though, the problem had not been around long.) Gerhard "Or Look Up 'On Hyperassociativity'" Paseman, 2019.07.26.
$endgroup$
– Gerhard Paseman
10 hours ago
add a comment |
2
$begingroup$
This happens frequently. Search for the word "independently" in abstracts or in the conclusion (or introduction) of papers. My name appears in Algebra Universalis because Libor Polak acknowledged my near simultaneous solution to a problem in logic he solved. (In this case though, the problem had not been around long.) Gerhard "Or Look Up 'On Hyperassociativity'" Paseman, 2019.07.26.
$endgroup$
– Gerhard Paseman
10 hours ago
2
2
$begingroup$
This happens frequently. Search for the word "independently" in abstracts or in the conclusion (or introduction) of papers. My name appears in Algebra Universalis because Libor Polak acknowledged my near simultaneous solution to a problem in logic he solved. (In this case though, the problem had not been around long.) Gerhard "Or Look Up 'On Hyperassociativity'" Paseman, 2019.07.26.
$endgroup$
– Gerhard Paseman
10 hours ago
$begingroup$
This happens frequently. Search for the word "independently" in abstracts or in the conclusion (or introduction) of papers. My name appears in Algebra Universalis because Libor Polak acknowledged my near simultaneous solution to a problem in logic he solved. (In this case though, the problem had not been around long.) Gerhard "Or Look Up 'On Hyperassociativity'" Paseman, 2019.07.26.
$endgroup$
– Gerhard Paseman
10 hours ago
add a comment |
10 Answers
10
active
oldest
votes
$begingroup$
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.
$endgroup$
5
$begingroup$
What a moment-ous occasion.
$endgroup$
– Arun Debray
8 hours ago
add a comment |
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
The Gelfond-Schneider Theorem, if $a$ and $b$ are algebraic numbers with $ane0,1$, and $b$ irrational, then any value of $a^b$ is a transcendental number, was proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
Richard Friedberg and Albert Muchnik independently and almost simultaneously solved Post's problem.
$endgroup$
add a comment |
$begingroup$
Ramanujan graphs were independently constructed by Margulis and by Lubotzky–Phillips–Sarnak.
$endgroup$
add a comment |
$begingroup$
The theory of NP-completeness was simultaneously and independently developed in the U.S. and the U.S.S.R.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f337023%2fexamples-of-simultaneous-independent-breakthroughs%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
10 Answers
10
active
oldest
votes
10 Answers
10
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
$endgroup$
5
$begingroup$
What a moment-ous occasion.
$endgroup$
– Arun Debray
8 hours ago
add a comment |
$begingroup$
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.
$endgroup$
5
$begingroup$
What a moment-ous occasion.
$endgroup$
– Arun Debray
8 hours ago
add a comment |
$begingroup$
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.
$endgroup$
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.
answered 10 hours ago
community wiki
Deane Yang
5
$begingroup$
What a moment-ous occasion.
$endgroup$
– Arun Debray
8 hours ago
add a comment |
5
$begingroup$
What a moment-ous occasion.
$endgroup$
– Arun Debray
8 hours ago
5
5
$begingroup$
What a moment-ous occasion.
$endgroup$
– Arun Debray
8 hours ago
$begingroup$
What a moment-ous occasion.
$endgroup$
– Arun Debray
8 hours ago
add a comment |
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
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
answered 10 hours ago
community wiki
Stanley Yao Xiao
add a comment |
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered 9 hours ago
community wiki
Alexandre Eremenko
add a comment |
add a comment |
$begingroup$
The Gelfond-Schneider Theorem, if $a$ and $b$ are algebraic numbers with $ane0,1$, and $b$ irrational, then any value of $a^b$ is a transcendental number, was proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
$endgroup$
add a comment |
$begingroup$
The Gelfond-Schneider Theorem, if $a$ and $b$ are algebraic numbers with $ane0,1$, and $b$ irrational, then any value of $a^b$ is a transcendental number, was proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
$endgroup$
add a comment |
$begingroup$
The Gelfond-Schneider Theorem, if $a$ and $b$ are algebraic numbers with $ane0,1$, and $b$ irrational, then any value of $a^b$ is a transcendental number, was proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
$endgroup$
The Gelfond-Schneider Theorem, if $a$ and $b$ are algebraic numbers with $ane0,1$, and $b$ irrational, then any value of $a^b$ is a transcendental number, was proved independently in 1934 by Aleksandr Gelfond and Theodor Schneider.
answered 5 hours ago
community wiki
Gerry Myerson
add a comment |
add a comment |
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
add a comment |
$begingroup$
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
$endgroup$
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
answered 10 hours ago
community wiki
Andreas Rüdinger
add a comment |
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered 4 hours ago
community wiki
bof
add a comment |
add a comment |
$begingroup$
Richard Friedberg and Albert Muchnik independently and almost simultaneously solved Post's problem.
$endgroup$
add a comment |
$begingroup$
Richard Friedberg and Albert Muchnik independently and almost simultaneously solved Post's problem.
$endgroup$
add a comment |
$begingroup$
Richard Friedberg and Albert Muchnik independently and almost simultaneously solved Post's problem.
$endgroup$
Richard Friedberg and Albert Muchnik independently and almost simultaneously solved Post's problem.
answered 4 hours ago
community wiki
Timothy Chow
add a comment |
add a comment |
$begingroup$
Ramanujan graphs were independently constructed by Margulis and by Lubotzky–Phillips–Sarnak.
$endgroup$
add a comment |
$begingroup$
Ramanujan graphs were independently constructed by Margulis and by Lubotzky–Phillips–Sarnak.
$endgroup$
add a comment |
$begingroup$
Ramanujan graphs were independently constructed by Margulis and by Lubotzky–Phillips–Sarnak.
$endgroup$
Ramanujan graphs were independently constructed by Margulis and by Lubotzky–Phillips–Sarnak.
answered 3 hours ago
community wiki
Timothy Chow
add a comment |
add a comment |
$begingroup$
The theory of NP-completeness was simultaneously and independently developed in the U.S. and the U.S.S.R.
$endgroup$
add a comment |
$begingroup$
The theory of NP-completeness was simultaneously and independently developed in the U.S. and the U.S.S.R.
$endgroup$
add a comment |
$begingroup$
The theory of NP-completeness was simultaneously and independently developed in the U.S. and the U.S.S.R.
$endgroup$
The theory of NP-completeness was simultaneously and independently developed in the U.S. and the U.S.S.R.
answered 3 hours ago
community wiki
Timothy Chow
add a comment |
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
add a comment |
$begingroup$
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.
$endgroup$
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.
answered 2 hours ago
community wiki
Yemon Choi
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f337023%2fexamples-of-simultaneous-independent-breakthroughs%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
$begingroup$
This happens frequently. Search for the word "independently" in abstracts or in the conclusion (or introduction) of papers. My name appears in Algebra Universalis because Libor Polak acknowledged my near simultaneous solution to a problem in logic he solved. (In this case though, the problem had not been around long.) Gerhard "Or Look Up 'On Hyperassociativity'" Paseman, 2019.07.26.
$endgroup$
– Gerhard Paseman
10 hours ago