{"id":13661,"date":"2023-09-14T11:38:11","date_gmt":"2023-09-14T15:38:11","guid":{"rendered":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13661"},"modified":"2023-09-27T17:54:59","modified_gmt":"2023-09-27T21:54:59","slug":"what-does-spec-z-look-like","status":"publish","type":"post","link":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/?p=13661","title":{"rendered":"What Does Spec Z<\/strong> Look Like?"},"content":{"rendered":"

This week Laurent Fargues has started a series of lectures here at Columbia on Some new geometric structures in the Langlands program<\/a>. Videos are available here<\/a>, but unfortunately there is a problem with the camera in that room, making the blackboard illegible (maybe we can get it fixed…). Fargues however is writing up detailed lecture notes, available here<\/a>, so you can follow along with those.<\/p>\n

Fargues is covering the story of the Fargues-Fontaine curve and the relationship between geometric Langlands on this curve and arithmetic local Langlands that he worked out with Scholze recently. On Monday Scholze gave a survey talk in Bonn entitled What Does Spec Z<\/strong> Look Like?<\/a>, video available here<\/a>. Scholze’s talk gave a speculative picture of how to think about the global arithmetic story, with Spec Z<\/strong> as a sort of three-dimensional space. One thing new to me was his picture of the real place as a puncture, with boundary the twistor projective line. He then went on to motivate the course he will be teaching this fall with Dustin Clausen on Analytic Stacks<\/a>. Here at Columbia we have an ongoing seminar on some of the background for this<\/a>, run by Juan Rodriguez Camargo and John Morgan.<\/p>\n

Update:<\/strong> Peter Scholze next week at the Rapoport conference will be giving a talk on new ideas about the twistor $$\\mathbf{P}^1$ and real local Langlands. His abstract is<\/p>\n

Towards a formulation of the real local Langlands correspondence as geometric Langlands on the twistor-$\\mathbf P^1$<\/p>\n

We will propose a formulation of the local Langlands correspondence for complex representations of real groups in terms of a(n everywhere unramified) geometric Langlands correspondence on the twistor-$\\mathbf P^1$, analogous to our work with Fargues in the case of p-adic groups. This is motivated by discussions with Rodriguez Camargo, Pan, le Bras and Ansch\u00fctz on the analogous case of locally analytic p-adic representations, and is different from the previous work of Ben-Zvi and Nadler in a similar direction. In particular, on the geometric side we get representations of the real group, encoded in terms of liquid quasicoherent sheaves on $[*\/G(\\mathbf R)^{la} ]$; and on the spectral side, we get representations of the real Weil group $W_R$, or rather vector bundles on $[(\\mathbf A^2\\backslash\\{0\\})\/W_R^{la} ]$.<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"

This week Laurent Fargues has started a series of lectures here at Columbia on Some new geometric structures in the Langlands program. Videos are available here, but unfortunately there is a problem with the camera in that room, making the … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[11],"tags":[],"class_list":["post-13661","post","type-post","status-publish","format-standard","hentry","category-langlands"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13661","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=13661"}],"version-history":[{"count":11,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13661\/revisions"}],"predecessor-version":[{"id":13683,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=\/wp\/v2\/posts\/13661\/revisions\/13683"}],"wp:attachment":[{"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=13661"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=13661"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.math.columbia.edu\/~woit\/wordpress\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=13661"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}