Fdtxjr

Is there a data structure that only stores hash codes and not the actual objects?

It's a yearly task, alright

Examples of the Pigeonhole Principle

Does Mathematica reuse previous computations?

Interplanetary conflict, some disease destroys the ability to understand or appreciate music

Time travel from stationary position?

The difference between「N分で」and「後N分で」

Brexit - No Deal Rejection

Why does Bach not break the rules here?

How to terminate ping <dest> &

What approach do we need to follow for projects without a test environment?

Are ETF trackers fundamentally better than individual stocks?

Identifying the interval from A♭ to D♯

A Cautionary Suggestion

Why doesn't the EU now just force the UK to choose between referendum and no-deal?

How to explain that I do not want to visit a country due to personal safety concern?

My adviser wants to be the first author

Does someone need to be connected to my network to sniff HTTP requests?

Opacity of an object in 2.8

How Could an Airship Be Repaired Mid-Flight

What are substitutions for coconut in curry?

Co-worker team leader wants to inject his friend's awful software into our development. What should I say to our common boss?

Existence of subset with given Hausdorff dimension

What is the significance behind "40 days" that often appears in the Bible?

What rigorous mathematical theorems has Edward Witten discovered?

What is mathematical equation of the Gravitational force between three objects?In what ways has physics spurred the invention of new mathematical tools?Guidance regarding research in Mathematical PhysicsWhat is a good reference for rigorous Electromagnetism and Electrodynamics?What is the mathematical understanding behind what physicists call a gauge fixing?A space more fundamental than Euclidean spaceA text that can accompany this Course on “Geometry for theoretical physics”What are the mathematical foundations of the renormalisation group?Was von Neumann's 1954 ICM address “Unsolved Problems in Mathematics” outdated?Intuitionism and theoretical physics

4

2

$begingroup$

I read that Ed Witten's 1990 Fields Medal was somewhat controversial among mathematicians, because even though no one questioned his deep conceptual understanding of important new mathematical ideas, his work was not considered rigorous enough to deserve the medal. For example, Wikipedia claims that

Witten's work [on Chern-Simons theory and topological quantum field theory] was based on the mathematically ill-defined notion of a Feynman path integral and was therefore not mathematically rigorous, [although] mathematicians were [later] able to systematically develop Witten's ideas.

Moreover, the actual discovery that his Fields Medal nominally awarded was an innovative new simpler proof of the positive energy theorem, which had already been proven.

What are some examples of theorems that Witten discovered that were not previously known, such that there is consensus across the mathematical community that (a) the hypothesis and conclusion of the theorem are completely and unambiguously mathematically well-posed and (b) the proof is completely mathematically rigorous?

Edit: I don't mean theorems that Witten proposed non-rigorously which were then later made rigorous by mathematicians (of which there are many). I mean theorems for which Witten himself provided the rigorous version. (I know there are eternal philosophical debates between mathematicians about the relative importance of imprecise conjectures vs. precise conjectures vs. rigorous proofs for statements that end up being correct. But presumably both correctness and proof are important for the kind of work that the Fields Medal's purpose is to recognize.)

mathematical-physics quantum-field-theory gauge-theory topological-quantum-field-theory

share|cite|improve this question

edited Mar 11 at 19:48

Andrews

1,2691421

asked Jun 1 '18 at 18:39

tparkertparker

1,926834

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy Style guides discourage using en dashes (used for sentence breaks or number ranges) rather than hyphens to join proper names together, so I rolled back your edit.
  $endgroup$
  – tparker
  Jun 1 '18 at 23:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which style guides? Wikipedia's manual of style requires that usage. So does the Pacific Journal of Mathematics.
  $endgroup$
  – Michael Hardy
  Jun 2 '18 at 2:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy The Chicago Manual of Style specifies that a hyphen should be used in the phrase "Michelson-Morley experiment", which is remarkably parallel to "Chern-Simons theory". Most online sources that I found recommend limiting en dashes to numerical ranges and a few more obscure cases; Wikipedia's style guidlines seem to be represent a small minority position.
  $endgroup$
  – tparker
  Jun 2 '18 at 3:44
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy But I don't feel strongly about it - if you want to roll back to v2 then I won't reroll back to v1.
  $endgroup$
  – tparker
  Jun 2 '18 at 3:46
  
  

  
  
  
  

add a comment | 

4

2

$begingroup$

I read that Ed Witten's 1990 Fields Medal was somewhat controversial among mathematicians, because even though no one questioned his deep conceptual understanding of important new mathematical ideas, his work was not considered rigorous enough to deserve the medal. For example, Wikipedia claims that

Witten's work [on Chern-Simons theory and topological quantum field theory] was based on the mathematically ill-defined notion of a Feynman path integral and was therefore not mathematically rigorous, [although] mathematicians were [later] able to systematically develop Witten's ideas.

Moreover, the actual discovery that his Fields Medal nominally awarded was an innovative new simpler proof of the positive energy theorem, which had already been proven.

What are some examples of theorems that Witten discovered that were not previously known, such that there is consensus across the mathematical community that (a) the hypothesis and conclusion of the theorem are completely and unambiguously mathematically well-posed and (b) the proof is completely mathematically rigorous?

Edit: I don't mean theorems that Witten proposed non-rigorously which were then later made rigorous by mathematicians (of which there are many). I mean theorems for which Witten himself provided the rigorous version. (I know there are eternal philosophical debates between mathematicians about the relative importance of imprecise conjectures vs. precise conjectures vs. rigorous proofs for statements that end up being correct. But presumably both correctness and proof are important for the kind of work that the Fields Medal's purpose is to recognize.)

mathematical-physics quantum-field-theory gauge-theory topological-quantum-field-theory

share|cite|improve this question

edited Mar 11 at 19:48

Andrews

1,2691421

asked Jun 1 '18 at 18:39

tparkertparker

1,926834

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy Style guides discourage using en dashes (used for sentence breaks or number ranges) rather than hyphens to join proper names together, so I rolled back your edit.
  $endgroup$
  – tparker
  Jun 1 '18 at 23:03
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Which style guides? Wikipedia's manual of style requires that usage. So does the Pacific Journal of Mathematics.
  $endgroup$
  – Michael Hardy
  Jun 2 '18 at 2:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy The Chicago Manual of Style specifies that a hyphen should be used in the phrase "Michelson-Morley experiment", which is remarkably parallel to "Chern-Simons theory". Most online sources that I found recommend limiting en dashes to numerical ranges and a few more obscure cases; Wikipedia's style guidlines seem to be represent a small minority position.
  $endgroup$
  – tparker
  Jun 2 '18 at 3:44
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @MichaelHardy But I don't feel strongly about it - if you want to roll back to v2 then I won't reroll back to v1.
  $endgroup$
  – tparker
  Jun 2 '18 at 3:46
  
  

  
  
  
  

add a comment | 

$begingroup$

I read that Ed Witten's 1990 Fields Medal was somewhat controversial among mathematicians, because even though no one questioned his deep conceptual understanding of important new mathematical ideas, his work was not considered rigorous enough to deserve the medal. For example, Wikipedia claims that

Witten's work [on Chern-Simons theory and topological quantum field theory] was based on the mathematically ill-defined notion of a Feynman path integral and was therefore not mathematically rigorous, [although] mathematicians were [later] able to systematically develop Witten's ideas.

Moreover, the actual discovery that his Fields Medal nominally awarded was an innovative new simpler proof of the positive energy theorem, which had already been proven.

What are some examples of theorems that Witten discovered that were not previously known, such that there is consensus across the mathematical community that (a) the hypothesis and conclusion of the theorem are completely and unambiguously mathematically well-posed and (b) the proof is completely mathematically rigorous?

Edit: I don't mean theorems that Witten proposed non-rigorously which were then later made rigorous by mathematicians (of which there are many). I mean theorems for which Witten himself provided the rigorous version. (I know there are eternal philosophical debates between mathematicians about the relative importance of imprecise conjectures vs. precise conjectures vs. rigorous proofs for statements that end up being correct. But presumably both correctness and proof are important for the kind of work that the Fields Medal's purpose is to recognize.)

mathematical-physics quantum-field-theory gauge-theory topological-quantum-field-theory

share|cite|improve this question

edited Mar 11 at 19:48

Andrews

1,2691421

asked Jun 1 '18 at 18:39

tparkertparker

1,926834

$endgroup$

share|cite|improve this question

edited Mar 11 at 19:48

Andrews

1,2691421

asked Jun 1 '18 at 18:39

tparkertparker

1,926834

share|cite|improve this question

edited Mar 11 at 19:48

Andrews

1,2691421

asked Jun 1 '18 at 18:39

tparkertparker

1,926834

edited Mar 11 at 19:48

Andrews

1,2691421

edited Mar 11 at 19:48

Andrews

1,2691421

asked Jun 1 '18 at 18:39

tparkertparker

1,926834

asked Jun 1 '18 at 18:39

tparkertparker

1,926834

1 Answer
1

active

oldest

votes

7

$begingroup$

Let a famous mathematician answer this question on the contributions of Edward Witten: On the work of Edward Witten, by Michael Atiyah.

share|cite|improve this answer

answered Jun 1 '18 at 18:45

Dietrich BurdeDietrich Burde

80.7k647104

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I don't think this really addresses my question because it doesn't specify the mathematical rigor of Witten's various results, which is the main point of my question. The article uses the word "rigorous" six times. Three of the uses specifically refer to (other) mathematicians' later formalizations of Witten's rough ideas. The other three are more ambiguous (e.g. "rigorous proofs ... have always been forthcoming"), but as far as I can tell do as well.
  $endgroup$
  – tparker
  Jun 1 '18 at 23:16
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I think, Atiyah clearly discusses and answers the question about the mathematical rigour. This is a main point to justify for awarding the Fields Medal in Mathematics.
  $endgroup$
  – Dietrich Burde
  Jun 2 '18 at 8:11
  

  
  
  
  

add a comment | 

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
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
);



);

draft saved

draft discarded

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

Name

Email

Required, but never shown

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2804666%2fwhat-rigorous-mathematical-theorems-has-edward-witten-discovered%23new-answer', 'question_page');

);

Post as a guest

Name

Email

Required, but never shown

7

$begingroup$

Let a famous mathematician answer this question on the contributions of Edward Witten: On the work of Edward Witten, by Michael Atiyah.

share|cite|improve this answer

answered Jun 1 '18 at 18:45

Dietrich BurdeDietrich Burde

80.7k647104

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I don't think this really addresses my question because it doesn't specify the mathematical rigor of Witten's various results, which is the main point of my question. The article uses the word "rigorous" six times. Three of the uses specifically refer to (other) mathematicians' later formalizations of Witten's rough ideas. The other three are more ambiguous (e.g. "rigorous proofs ... have always been forthcoming"), but as far as I can tell do as well.
  $endgroup$
  – tparker
  Jun 1 '18 at 23:16
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I think, Atiyah clearly discusses and answers the question about the mathematical rigour. This is a main point to justify for awarding the Fields Medal in Mathematics.
  $endgroup$
  – Dietrich Burde
  Jun 2 '18 at 8:11
  

  
  
  
  

add a comment | 

7

$begingroup$

Let a famous mathematician answer this question on the contributions of Edward Witten: On the work of Edward Witten, by Michael Atiyah.

share|cite|improve this answer

answered Jun 1 '18 at 18:45

Dietrich BurdeDietrich Burde

80.7k647104

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I don't think this really addresses my question because it doesn't specify the mathematical rigor of Witten's various results, which is the main point of my question. The article uses the word "rigorous" six times. Three of the uses specifically refer to (other) mathematicians' later formalizations of Witten's rough ideas. The other three are more ambiguous (e.g. "rigorous proofs ... have always been forthcoming"), but as far as I can tell do as well.
  $endgroup$
  – tparker
  Jun 1 '18 at 23:16
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  I think, Atiyah clearly discusses and answers the question about the mathematical rigour. This is a main point to justify for awarding the Fields Medal in Mathematics.
  $endgroup$
  – Dietrich Burde
  Jun 2 '18 at 8:11
  

  
  
  
  

add a comment | 

$begingroup$

Let a famous mathematician answer this question on the contributions of Edward Witten: On the work of Edward Witten, by Michael Atiyah.

share|cite|improve this answer

answered Jun 1 '18 at 18:45

Dietrich BurdeDietrich Burde

80.7k647104

$endgroup$

share|cite|improve this answer

answered Jun 1 '18 at 18:45

Dietrich BurdeDietrich Burde

80.7k647104

answered Jun 1 '18 at 18:45

Dietrich BurdeDietrich Burde

80.7k647104

answered Jun 1 '18 at 18:45

Dietrich BurdeDietrich Burde

80.7k647104