Editing Alain Connes (section)

== Research ==
Connes' main research interests revolved around [[operator algebra]]s. Besides [[noncommutative geometry]], he has applied his works in various areas of mathematics and [[number theory]], [[differential geometry]].<ref name=":0" /><ref>Alexander Hellemans, [http://www.scientificamerican.com/article.cfm?id=the-geometer-of-particle "The Geometer of Particle Physics"] '' Scientific American'', 24 July 2006</ref><ref name=":1" /><ref name=":2" /> Since the 1990s, he developed [[noncommutative geometry]].<ref>{{cite book | author = [[Sabine Hossenfelder]] | title = Lost in Math: How Beauty Leads Physics Astray | isbn=978-0465094257 | year = 2018| publisher = Basic Books }}</ref>

In his early work on [[von Neumann algebra]]s in the 1970s, he succeeded in obtaining the almost complete classification of injective [[Von Neumann algebra#Factors|factor]]s. He also formulated the [[Connes embedding problem]].

Following this, he made contributions in [[K-theory|operator K-theory]] and [[index theory]], which culminated in the [[Baum–Connes conjecture]]. He also introduced [[cyclic homology|cyclic cohomology]] in the early 1980s as a first step in the study of [[noncommutative geometry|noncommutative differential geometry]].<ref>{{Cite journal |last=Moore |first=Calvin |author-link=Calvin C. Moore |date=1982 |title=The work of Alain Connes |url=https://www.ams.org/journals/notices/198210/198210FullIssue.pdf?cat=fullissue&trk=fullissue198210 |journal=[[Notices of the American Mathematical Society]] |volume=29 |issue=6 |pages=499–501}}</ref><ref>{{Cite journal |last=Seda |first=Anthony Karel |date=1984 |title=One aspect of the work of Alain Connes |url=https://www.irishmathsoc.org/nl11/nl11_38-48.pdf |journal=[[Irish Mathematical Society#Bulletin of the Irish Mathematical Society|Irish Mathematical Society Newsletter]] |volume=11 |pages=38–48|doi=10.33232/BIMS.0011.38.48 }}</ref><ref>{{Cite journal |last=Lesniewski |first=Andrew |date=1997 |title=Noncommutative Geometry |url=https://www.ams.org/notices/199707/lesniewski.pdf |journal=[[Notices of the American Mathematical Society]] |volume=44 |issue=7 |pages=800–805}}</ref><ref>{{Cite journal |last1=Cornelissen |first1=Gunther |last2=Landsman |first2=Klaas |last3=van Suijlekom |first3=Walter |date=2010 |title=The flashes of insight never came for free |url=https://www.nieuwarchief.nl/serie5/pdf/naw5-2010-11-4-250.pdf |journal=[[Nieuw Archief voor Wiskunde]] |volume=11 |issue=5 |pages=250–256}}</ref>

He was a member of [[Nicolas Bourbaki]].<ref>{{cite book |title=Bourbaki: a secret society of mathematicians|first=Maurice|last=Mashaal|publisher=[[American Mathematical Society]]|year=2006|page=18|isbn=978-0-8218-3967-6|url=https://books.google.com/books?id=-CXn6y_1nJ8C&pg=PA18}}</ref> Over many years, he collaborated extensively with [[Henri Moscovici]].<ref name=":0"/>