Editing Michael Atiyah (section)

===Later work (1986–2019)===
[[File:Edward Witten at Harvard.jpg|thumb|right|[[Edward Witten]], whose work on invariants of manifolds and [[topological quantum field theories]] was influenced by Atiyah]]
Many of the papers in the 6th volume<ref>{{harvs|nb|first=Michael|last=Atiyah|year1=2004}}</ref> of his collected works are surveys, obituaries, and general talks. Atiyah continued to publish subsequently, including several surveys, a popular book,<ref>{{harvnb|Atiyah|2007}}</ref> and another paper with [[Graeme Segal|Segal]] on [[twisted K-theory]].

One paper<ref>{{harvnb|Atiyah|2004|loc=paper 127}}</ref> is a detailed study of the [[Dedekind eta function]] from the point of view of topology and the index theorem.

Several of his papers from around this time study the connections between [[quantum field theory]], [[knot theory|knots]], and [[Donaldson theory]]. He introduced the concept of a [[topological quantum field theory]], inspired by Witten's work and Segal's definition of a conformal field theory.<ref>{{harvnb|Atiyah|2004|loc=paper 132}}</ref> His book "The Geometry and Physics of Knots"<ref>{{harvnb|Atiyah|1990}}</ref> describes the new [[knot invariant]]s found by [[Vaughan Jones]] and [[Edward Witten]] in terms of topological quantum field theories, and his paper with L. Jeffrey<ref>{{harvnb|Atiyah|2004|loc=paper 139}}</ref> explains Witten's Lagrangian giving the [[Donaldson invariant]]s.

He studied [[skyrmion]]s with Nick Manton,<ref>{{harvnb|Atiyah|2004|loc=papers 141, 142}}</ref> finding a relation with [[magnetic monopoles]] and [[instanton]]s, and giving a conjecture for the structure of the [[moduli space]] of two [[skyrmions]] as a certain [[subquotient]] of complex [[projective 3-space]].

Several papers<ref>{{harvnb|Atiyah|2004|loc=papers 163, 164, 165, 166, 167, 168}}</ref> were inspired by a question of [https://research-information.bristol.ac.uk/en/persons/jonathan-m-robbins(60101278-f877-4e14-946f-262f3b95d5ae).html Jonathan Robbins] (called the [[Berry–Robbins problem]]), who asked if there is a map from the configuration space of ''n'' points in 3-space to the flag manifold of the unitary group. Atiyah gave an affirmative answer to this question, but felt his solution was too computational and studied a conjecture that would give a more natural solution. He also related the question to [[Nahm's equation]], and introduced the [[Atiyah conjecture on configurations]].

{{quote box
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|quote=But for most practical purposes, you just use the classical groups. The exceptional Lie groups are just there to show you that the theory is a bit bigger; it is pretty rare that they ever turn up.
|source=Michael Atiyah<ref name="ReferenceA">{{harvnb|Atiyah|1988a|loc = paper 19, p. 19}}</ref>
}}
With [[Juan Maldacena]] and [[Cumrun Vafa]],<ref>{{harvnb|Atiyah|2004|loc=paper 169}}</ref> and [[E. Witten]]<ref>{{harvnb|Atiyah|2004|loc=paper 170}}</ref> he described the dynamics of [[M-theory]] on [[Joyce manifold|manifolds with G<sub>2</sub> holonomy]]. These papers seem to be the first time that Atiyah worked on exceptional Lie groups.

In his papers with [[Michael J. Hopkins|M. Hopkins]]<ref>{{harvnb|Atiyah|2004|loc=paper 172}}</ref> and G. Segal<ref>{{harvnb|Atiyah|2004|loc=paper 173}}</ref> he returned to his earlier interest of K-theory, describing some twisted forms of K-theory with applications in [[theoretical physics]].

In October 2016, he claimed<ref>{{cite arXiv |last=Atiyah |first=Michael |eprint=1610.09366 |title=The Non-Existent Complex 6-Sphere|class=math.DG |year=2016 }}</ref> a short proof of the non-existence of [[Complex manifold|complex structures]] on the 6-sphere. His proof, like many predecessors, is considered flawed by the mathematical community, even after the proof was rewritten in a revised form.<ref>{{citation|url=https://mathoverflow.net/q/263301|title=What is the current understanding regarding complex structures on the 6-sphere? (MathOverflow)|access-date=24 September 2018 }}</ref><ref>{{citation|url=https://mathoverflow.net/q/304071 |title=Atiyah's May 2018 paper on the 6-sphere (MathOverflow)|access-date=24 September 2018 }}</ref>

At the 2018 [[Klaus Tschira Foundation|Heidelberg Laureate Forum]], he claimed to have solved the [[Riemann hypothesis]], [[Hilbert's eighth problem]], [[proof by contradiction|by contradiction]] using the [[fine-structure constant]]. Again, the proof did not hold up and the hypothesis remains one of the six unsolved [[Millennium Prize Problems]] in mathematics, as of 2025.<ref>{{Cite news|url=https://www.science.org/content/article/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis|title=Skepticism surrounds renowned mathematician's attempted proof of 160-year-old hypothesis|date=24 September 2018|work=Science {{!}} AAAS|access-date=26 September 2018|language=en|archive-url=https://web.archive.org/web/20180926115652/https://www.sciencemag.org/news/2018/09/skepticism-surrounds-renowned-mathematician-s-attempted-proof-160-year-old-hypothesis|archive-date=26 September 2018|url-status=live}}</ref><ref>{{cite news|url=https://www.newscientist.com/article/2180504-riemann-hypothesis-likely-remains-unsolved-despite-claimed-proof/|title=Riemann hypothesis likely remains unsolved despite claimed proof|access-date=24 September 2018|archive-url=https://web.archive.org/web/20180924175329/https://www.newscientist.com/article/2180504-riemann-hypothesis-likely-remains-unsolved-despite-claimed-proof/|archive-date=24 September 2018|url-status=live}}</ref>