Editing Edward Witten (section)

===Fields medal work===
Witten was awarded the [[Fields Medal]] by the [[International Mathematical Union]] in 1990.<ref>{{Cite web|date=2011|title=Edward Witten|url=http://www.sns.ias.edu/~witten/CurrentCV.pdf|url-status=dead|archive-url=https://web.archive.org/web/20120204111241/http://www.sns.ias.edu/~witten/CurrentCV.pdf|archive-date=February 4, 2012|access-date=April 13, 2021}}</ref>

In a written address to the [[International Congress of Mathematicians|ICM]], [[Michael Atiyah]] said of Witten:<ref name="atiyah" />

{{blockquote|text=Although he is definitely a physicist (as his list of publications clearly shows) his command of mathematics is rivaled by few mathematicians, and his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems&nbsp;... He has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics.<ref name="atiyah" />}}

{{stack|[[File:Widden Mori.jpg|thumb|Edward Witten (left) with mathematician [[Shigefumi Mori]], probably at the [[International Congress of Mathematicians|ICM]] in 1990, where they received the [[Fields Medal]]]]}}
As an example of Witten's work in pure mathematics, Atiyah cites his application of techniques from [[quantum field theory]] to the mathematical subject of [[low-dimensional topology]]. In the late 1980s, Witten coined the term ''[[topological quantum field theory]]'' for a certain type of physical theory in which the [[expectation value]]s of observable quantities encode information about the [[topology]] of [[spacetime]].<ref name="tqft">{{Citation | last1=Witten | first1=Edward | year=1988 | title=Topological quantum field theory | url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1104161738 | journal=[[Communications in Mathematical Physics]] | volume=117 | issue=3 | pages=353–386 |bibcode = 1988CMaPh.117..353W |doi = 10.1007/BF01223371 | s2cid=43230714 }}</ref> In particular, Witten realized that a physical theory now called [[Chern–Simons theory]] could provide a framework for understanding the mathematical theory of [[knot (mathematics)|knots]] and [[3-manifold]]s.<ref>{{Cite journal |last=Witten |first=Edward |year=1989 |title=Quantum Field Theory and the Jones Polynomial |url=http://www.maths.ed.ac.uk/~aar/papers/witten.pdf |journal=[[Communications in Mathematical Physics]] |volume=121 |issue=3 |pages=351–399 |bibcode = 1989CMaPh.121..351W |doi = 10.1007/BF01217730 |s2cid=14951363 }}</ref> Although Witten's work was based on the mathematically ill-defined notion of a [[Feynman path integral]] and therefore not [[mathematical rigor|mathematically rigorous]], mathematicians were able to systematically develop Witten's ideas, leading to the theory of [[Reshetikhin–Turaev invariant]]s.<ref>{{cite journal |last1=Reshetikhin |first1=Nicolai |last2=Turaev |first2=Vladimir |year=1991 |title=Invariants of 3-manifolds via link polynomials and quantum groups |journal=[[Inventiones Mathematicae]] |volume=103 |issue=1 |pages=547–597 |bibcode = 1991InMat.103..547R |doi = 10.1007/BF01239527 |s2cid=123376541 }}</ref>

Another result for which Witten was awarded the Fields Medal was his proof in 1981 of the [[positive energy theorem]] in [[general relativity]].<ref>{{cite journal |last1=Witten |first1=Edward |year=1981 |title=A new proof of the positive energy theorem |url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103919981 |journal=[[Communications in Mathematical Physics]] |volume=80 |issue=3 |pages=381–402 |bibcode = 1981CMaPh..80..381W |doi = 10.1007/BF01208277 |s2cid=1035111 }}</ref> This theorem asserts that (under appropriate assumptions) the total [[energy]] of a gravitating system is always positive and can be zero only if the geometry of [[spacetime]] is that of flat [[Minkowski space]]. It establishes Minkowski space as a stable ground state of the [[gravitational field]]. While the original proof of this result due to [[Richard Schoen]] and [[Shing-Tung Yau]] used [[variational methods]],<ref>{{cite journal |last1=Schoen |first1=Robert |last2=Yau |first2=Shing-Tung |year=1979 |title=On the proof of the positive mass conjecture in general relativity |journal=[[Communications in Mathematical Physics]] |volume=65 |issue=1 |page=45 |bibcode = 1979CMaPh..65...45S |doi = 10.1007/BF01940959 |s2cid=54217085 |url=http://projecteuclid.org/euclid.cmp/1103904790 }}</ref><ref>{{cite journal |last1=Schoen |first1=Robert |last2=Yau |first2=Shing-Tung |year=1981 |title=Proof of the positive mass theorem. II |journal=[[Communications in Mathematical Physics]] |volume=79 |issue=2 |page=231 |bibcode = 1981CMaPh..79..231S |doi = 10.1007/BF01942062 |s2cid=59473203 |url=http://projecteuclid.org/euclid.cmp/1103908964 }}</ref> Witten's proof used ideas from [[supergravity theory]] to simplify the argument.<ref name="Parker 1985 Gauge choice in Witten">{{cite journal | last=Parker | first=Thomas H. | title=Gauge choice in Witten's energy expression | journal=Communications in Mathematical Physics | volume=100 | issue=4 | date=1985 | issn=0010-3616 | doi=10.1007/BF01217725 | pages=471–480| bibcode=1985CMaPh.100..471P | url=http://projecteuclid.org/euclid.cmp/1104114001 }}</ref>

A third area mentioned in Atiyah's address is Witten's work relating [[supersymmetry]] and [[Morse theory]],<ref name="Witten 1982 Supersymmetry and Morse theory">{{cite journal | last1=Witten | first1=Edward | title=Super-symmetry and Morse Theory | year=1982 | journal=[[Journal of Differential Geometry]]| pages=661–692 | volume=17| issue=4 | doi=10.4310/jdg/1214437492 | doi-access=free }}</ref> a branch of mathematics that studies the [[topology]] of [[manifolds]] using the concept of a [[differentiable function]]. Witten's work gave a physical proof of a classical result, the [[Morse theory#Morse inequalities|Morse inequalities]], by interpreting the theory in terms of [[supersymmetric quantum mechanics]].<ref name="Witten 1982 Supersymmetry and Morse theory"/>