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mw-first-heading"><span class="mw-page-title-main">Jean-Louis Verdier</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">French mathematician</div> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox biography vcard"><tbody><tr><th colspan="2" class="infobox-above" style="font-size:125%;"><div class="fn">Jean-Louis Verdier</div></th></tr><tr><td colspan="2" class="infobox-image"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/File:Verdier_Previato.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Verdier_Previato.jpg/220px-Verdier_Previato.jpg" decoding="async" width="220" height="322" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Verdier_Previato.jpg/330px-Verdier_Previato.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f4/Verdier_Previato.jpg/440px-Verdier_Previato.jpg 2x" data-file-width="544" data-file-height="797" /></a></span><div class="infobox-caption">Jean-Louis Verdier (right) and <a href="/wiki/Emma_Previato" title="Emma Previato">Emma Previato</a>, <a href="/wiki/Mathematical_Research_Institute_of_Oberwolfach" class="mw-redirect" title="Mathematical Research Institute of Oberwolfach">Oberwolfach</a> 1984</div></td></tr><tr><th scope="row" class="infobox-label">Born</th><td class="infobox-data"><span style="display:none">(<span class="bday">1935-02-02</span>)</span>2 February 1935</td></tr><tr><th scope="row" class="infobox-label">Died</th><td class="infobox-data">25 August 1989<span style="display:none">(1989-08-25)</span> (aged 54)</td></tr><tr><th scope="row" class="infobox-label">Nationality</th><td class="infobox-data category">French</td></tr><tr><th scope="row" class="infobox-label">Alma mater</th><td class="infobox-data"><a href="/wiki/University_of_Paris" title="University of Paris">University of Paris</a></td></tr><tr><th scope="row" class="infobox-label">Known for</th><td class="infobox-data"><a href="/wiki/Verdier_duality" title="Verdier duality">Verdier duality</a><br /><span class="nowrap"><a href="/wiki/Hypercovering" title="Hypercovering">Verdier hypercovering theorem</a></span><br /><a href="/wiki/Artin%E2%80%93Verdier_duality" title="Artin–Verdier duality">Artin–Verdier duality</a><br /><span class="nowrap"><a href="/wiki/Coherent_duality" title="Coherent duality">Serre–Grothendieck–Verdier duality</a></span><br /><a href="/wiki/Derived_category" title="Derived category">Derived category</a><br /><a href="/wiki/Triangulated_category" title="Triangulated category">Triangulated category</a></td></tr><tr><td colspan="2" class="infobox-full-data"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1257001546"><b>Scientific career</b></td></tr><tr><th scope="row" class="infobox-label">Fields</th><td class="infobox-data category"><a href="/wiki/Mathematics" title="Mathematics">Mathematics</a></td></tr><tr><th scope="row" class="infobox-label">Institutions</th><td class="infobox-data"><a href="/wiki/Paris_Diderot_University" title="Paris Diderot University">Paris Diderot University</a></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Thesis" title="Thesis">Thesis</a></th><td class="infobox-data"><i><a rel="nofollow" class="external text" href="http://www.numdam.org/item/AST_1996__239__R1_0.pdf">Des catégories dérivées des categories abéliennes</a></i> <span style="font-size:97%;">(1967)</span></td></tr><tr><th scope="row" class="infobox-label"><a href="/wiki/Doctoral_advisor" title="Doctoral advisor">Doctoral advisor</a></th><td class="infobox-data"><a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a></td></tr><tr><th scope="row" class="infobox-label">Doctoral students</th><td class="infobox-data"><a href="/wiki/Arnaud_Beauville" title="Arnaud Beauville">Arnaud Beauville</a><br /><a href="/wiki/Alain_Lascoux" title="Alain Lascoux">Alain Lascoux</a></td></tr><tr style="display:none"><td colspan="2"> </td></tr></tbody></table> <p><b>Jean-Louis Verdier</b> (<style data-mw-deduplicate="TemplateStyles:r1177148991">.mw-parser-output .IPA-label-small{font-size:85%}.mw-parser-output .references .IPA-label-small,.mw-parser-output .infobox .IPA-label-small,.mw-parser-output .navbox .IPA-label-small{font-size:100%}</style><span class="IPA-label IPA-label-small">French:</span> <span class="IPA nowrap" lang="fr-Latn-fonipa"><a href="/wiki/Help:IPA/French" title="Help:IPA/French">[vɛʁdje]</a></span>; 2 February 1935 – 25 August 1989) was a <a href="/wiki/French_people" title="French people">French</a> <a href="/wiki/Mathematician" title="Mathematician">mathematician</a> who worked, under the guidance of his doctoral advisor <a href="/wiki/Alexander_Grothendieck" title="Alexander Grothendieck">Alexander Grothendieck</a>, on <a href="/wiki/Derived_category" title="Derived category">derived categories</a> and <a href="/wiki/Verdier_duality" title="Verdier duality">Verdier duality</a>. He was a close collaborator of Grothendieck, notably contributing to <a href="/wiki/SGA_4" class="mw-redirect" title="SGA 4">SGA 4</a> his theory of <a href="/w/index.php?title=Hypercovers&action=edit&redlink=1" class="new" title="Hypercovers (page does not exist)">hypercovers</a> and anticipating the later development of <a href="/wiki/%C3%89tale_homotopy" class="mw-redirect" title="Étale homotopy">étale homotopy</a> by <a href="/wiki/Michael_Artin" title="Michael Artin">Michael Artin</a> and <a href="/wiki/Barry_Mazur" title="Barry Mazur">Barry Mazur</a>, following a suggestion he attributed to <a href="/wiki/Pierre_Cartier_(mathematician)" title="Pierre Cartier (mathematician)">Pierre Cartier</a>. <a href="/w/index.php?title=Saul_Lubkin&action=edit&redlink=1" class="new" title="Saul Lubkin (page does not exist)">Saul Lubkin</a>'s related theory of <a href="/w/index.php?title=Rigid_hypercover&action=edit&redlink=1" class="new" title="Rigid hypercover (page does not exist)">rigid hypercovers</a> was later taken up by <a href="/wiki/Eric_Friedlander" title="Eric Friedlander">Eric Friedlander</a> in his definition of the <a href="/w/index.php?title=%C3%89tale_topological_type&action=edit&redlink=1" class="new" title="Étale topological type (page does not exist)">étale topological type</a>. </p><p>Verdier was a student at the elite <a href="/wiki/%C3%89cole_Normale_Sup%C3%A9rieure" class="mw-redirect" title="École Normale Supérieure">École Normale Supérieure</a> in Paris, and later became director of studies there, as well as a Professor at the <a href="/wiki/University_of_Paris-VII" class="mw-redirect" title="University of Paris-VII">University of Paris VII</a>. For many years he directed a joint seminar at the École Normale Supérieure with <a href="/wiki/Adrien_Douady" title="Adrien Douady">Adrien Douady</a>. Verdier was a member of <a href="/wiki/Nicolas_Bourbaki" title="Nicolas Bourbaki">Bourbaki</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In 1984 he was the president of the <a href="/wiki/Soci%C3%A9t%C3%A9_Math%C3%A9matique_de_France" class="mw-redirect" title="Société Mathématique de France">Société Mathématique de France</a>. </p><p>In 1976 Verdier developed a useful regularity condition on <a href="/w/index.php?title=Stratified_sets&action=edit&redlink=1" class="new" title="Stratified sets (page does not exist)">stratified sets</a> that the Chinese-Australian mathematician <a href="/w/index.php?title=Tzee-Char_Kuo&action=edit&redlink=1" class="new" title="Tzee-Char Kuo (page does not exist)">Tzee-Char Kuo</a> had previously shown implied the <a href="/wiki/Whitney_conditions" title="Whitney conditions">Whitney conditions</a> for <a href="/wiki/Subanalytic_set" title="Subanalytic set">subanalytic sets</a> (such as real or complex analytic varieties). Verdier called the condition (w) for <a href="/wiki/Hassler_Whitney" title="Hassler Whitney">Whitney</a>, as at the time he thought (w) might be equivalent to Whitney's condition (b). Real algebraic examples for which the Whitney conditions (b) hold but Verdier's condition (w) fails, were constructed by <a href="/wiki/David_Trotman" title="David Trotman">David Trotman</a> who has obtained many geometric properties of (w)-regular stratifications. Work of Bernard Teissier, aided by Jean-Pierre Henry and Michel Merle at the <a href="/wiki/%C3%89cole_Polytechnique" class="mw-redirect" title="École Polytechnique">École Polytechnique</a>, led to the 1982 result that Verdier's condition (w) is equivalent to the Whitney conditions for complex analytic stratifications. </p><p>Verdier later worked on the theory of <a href="/wiki/Integrable_system" title="Integrable system">integrable systems</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jean-Louis_Verdier&action=edit&section=1" title="Edit section: Bibliography"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Verdier's 1967 thesis, published belatedly in: <dl><dd><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFVerdier1996" class="citation journal cs1 cs1-prop-foreign-lang-source">Verdier, Jean-Louis (1996). "Des Catégories Dérivées des Catégories Abéliennes". <i>Astérisque</i> (in French). <b>239</b>. Société Mathématique de France, Marseilles.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ast%C3%A9risque&rft.atitle=Des+Cat%C3%A9gories+D%C3%A9riv%C3%A9es+des+Cat%C3%A9gories+Ab%C3%A9liennes&rft.volume=239&rft.date=1996&rft.aulast=Verdier&rft.aufirst=Jean-Louis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJean-Louis+Verdier" class="Z3988"></span></dd></dl></li></ul> <dl><dd>Part of it also appears in <a href="/wiki/Grothendieck%27s_S%C3%A9minaire_de_g%C3%A9om%C3%A9trie_alg%C3%A9brique#Bibliographic_information" class="mw-redirect" title="Grothendieck's Séminaire de géométrie algébrique">SGA 4½</a> as the last chapter, "Catégories dérivées (état 0)".</dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFVerdier1976" class="citation journal cs1">Verdier, Jean-Louis (1976). "Stratifications de Whitney et théorème de Bertini-Sard". <i><a href="/wiki/Inventiones_Mathematicae" title="Inventiones Mathematicae">Inventiones Mathematicae</a></i>. <b>36</b>: 295–312. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1976InMat..36..295V">1976InMat..36..295V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01390015">10.1007/BF01390015</a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0481096">0481096</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:118086083">118086083</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Inventiones+Mathematicae&rft.atitle=Stratifications+de+Whitney+et+th%C3%A9or%C3%A8me+de+Bertini-Sard&rft.volume=36&rft.pages=295-312&rft.date=1976&rft_id=info%3Adoi%2F10.1007%2FBF01390015&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D0481096%23id-name%3DMR&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A118086083%23id-name%3DS2CID&rft_id=info%3Abibcode%2F1976InMat..36..295V&rft.aulast=Verdier&rft.aufirst=Jean-Louis&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJean-Louis+Verdier" class="Z3988"></span></li> <li><a rel="nofollow" class="external text" href="https://www.springer.com/birkhauser/mathematics/book/978-0-8176-3653-1"><i>Integrable Systems, The Verdier Memorial Conference</i></a> (Actes du Colloque International de Luminy, 1991), Progress in Mathematics 115, edited by O. Babelon, P. Cartier, <a href="/wiki/Yvette_Kosmann-Schwarzbach" title="Yvette Kosmann-Schwarzbach">Y. Kosmann-Schwarzbach</a>, Birkhäuser, 1993.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jean-Louis_Verdier&action=edit&section=2" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Artin%E2%80%93Verdier_duality" title="Artin–Verdier duality">Artin–Verdier duality</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jean-Louis_Verdier&action=edit&section=3" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMashaal2006" class="citation cs2">Mashaal, Maurice (2006), <a rel="nofollow" class="external text" href="https://books.google.com/books?id=-CXn6y_1nJ8C"><i>Bourbaki: a secret society of mathematicians</i></a>, <a href="/wiki/American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8218-3967-6" title="Special:BookSources/978-0-8218-3967-6"><bdi>978-0-8218-3967-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Bourbaki%3A+a+secret+society+of+mathematicians&rft.pub=American+Mathematical+Society&rft.date=2006&rft.isbn=978-0-8218-3967-6&rft.aulast=Mashaal&rft.aufirst=Maurice&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D-CXn6y_1nJ8C&rfr_id=info%3Asid%2Fen.wikipedia.org%3AJean-Louis+Verdier" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Olivier Babelon, <a href="/wiki/Pierre_Cartier_(mathematician)" title="Pierre Cartier (mathematician)">Pierre Cartier</a>, <a href="/wiki/Yvette_Kosmann-Schwarzbach" title="Yvette Kosmann-Schwarzbach">Yvette Kosmann-Schwarzbach</a>: <i>Integrable systems. The Verdier memorial colloquium.</i> Birkhäuser, Progress in Mathematics, 1993.</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Jean-Louis_Verdier&action=edit&section=4" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://mathgenealogy.org/id.php?id=31267">Jean-Louis Verdier</a> at the <a href="/wiki/Mathematics_Genealogy_Project" title="Mathematics Genealogy Project">Mathematics Genealogy Project</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist 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