{"id":154,"date":"2023-06-16T14:22:17","date_gmt":"2023-06-16T21:22:17","guid":{"rendered":"https:\/\/live-usc-dornsife.pantheonsite.io\/aravind-asok\/?page_id=154"},"modified":"2026-04-20T11:00:53","modified_gmt":"2026-04-20T18:00:53","slug":"notes-lectures-informal-writings","status":"publish","type":"page","link":"https:\/\/dornsife.usc.edu\/aravind-asok\/notes-lectures-informal-writings\/","title":{"rendered":"Notes, Lectures, Informal Writings"},"content":{"rendered":"\n\n  \n    \n\n\n\n\n\n\n<div\n  class=\"cc--component-container cc--rich-text \"\n\n  \n  \n  \n  \n  \n  \n  >\n  <div class=\"c--component c--rich-text\"\n    \n      >\n\n    \n      \n<div class=\"f--field f--wysiwyg\">\n\n    \n  <h2><strong>Videos<\/strong><\/h2>\n<ul>\n<li><em><strong>Vector bundles on algebraic varieties<\/strong><\/em>\u00a0(with Jean Fasel); ICM 2022 [<a href=\"https:\/\/www.youtube.com\/watch?v=VJbG6MP2UGg\">YouTube<\/a>]<\/li>\n<li><em><strong>Vector bundles and A^1-homotopy theory<\/strong><\/em>; 2015 AMS Summer Algebraic Geometry Program [<a href=\"https:\/\/www.youtube.com\/watch?v=HiSBMzPDxVw\">YouTube<\/a>]<\/li>\n<\/ul>\n<h2><strong>Extended abstracts<\/strong><\/h2>\n<ul>\n<li><strong><i>Rational points vs. 0-cycles of degree 1 in stable A^1-homotopy<\/i><\/strong>; with C. Haesemeyer and F. Morel<br \/>\n(v. 16 Jun 2010) Oberwolfach Reports Volume 7, Issue 2, 2010 pp. 1423-1425\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/RatPtsOberwolfach.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>Toward a metastable range in A^1-homotopy theory of punctured affine spaces<\/i><\/strong>; with J. Fasel<br \/>\n(v. 22 Jul 2013) Oberwolfach Reports Volume 10, Issue 2, 2013 pp. 1892-1895\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/pinanminus0OWR.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong>On the Freudenthal suspension theorem in unstable motivic homotopy theory<\/strong>; with T. Bachmann and M.J. Hopkins (v. 1 Jun 2022) Oberwolfach Reports 2022\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/FreudenthalOWR.pdf\">[PDF]<\/a><\/li>\n<\/ul>\n<h2><strong>Lecture Notes<\/strong><\/h2>\n<ul>\n<li><strong><i>A^1-contractibility and topological contractibility<\/i><\/strong>\u00a0&#8211;\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/OttawaLectures.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><br \/>\nNotes for lectures at the Fields Institute workshop entitled &#8220;Group actions, generalized cohomology theories and affine algebraic geometry&#8221; at U. Ottawa<\/p>\n<ul>\n<li>An affirmative answer to Question 2.4.5 was given by A. Dubouloz and J. Fasel in their paper\u00a0<a href=\"https:\/\/arxiv.org\/abs\/1512.01933\">Families of A^1-contractible affine threefolds<\/a>. The work of Dubouloz and Fasel was motivated in part by the work of M. Hoyois, A. Krishna and P.-A. Ostvaer entitled\u00a0<a href=\"https:\/\/arxiv.org\/abs\/1409.1293\">A^1-contractibility of Koras-Russell threefolds<\/a>which establishes, among other things, that the Russell cubic is stably A^1-contractible; the latter also demolishes 2.4.8 (Approach 2) of the notes.<\/li>\n<li>To the best of my knowledge, Conjecture 2.3.9 is still open, even though claims to the contrary exist in, e.g., my Duke paper with Doran.<\/li>\n<li>(Update: 12\/21\/2021) Further advances can be found in\u00a0<a href=\"https:\/\/doi.org\/10.1007\/978-3-030-78977-0_5\">A^1-homotopy theory and contractible varieties: a survey<\/a>\u00a0written jointly with P.-A. Ostvaer.<\/li>\n<\/ul>\n<\/li>\n<li><strong><i>Algebraic geometry from an A^1-homotopic viewpoint<\/i><\/strong>\u00a0&#8211;\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/10\/MATH599.S16.pdf\" target=\"_blank\" rel=\"noopener\">[PDF 2016 version]<\/a>\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/MATH614.S21.pdf\">[PDF 2021 version]<\/a><br \/>\nNotes from a course at USC that studies naive A^1-homotopy theory. One main goal is to prove that the set of isomorphism classes of rank r vector bundles on a smooth affine scheme is represented by a suitable Grassmannian in the &#8220;naive A^1-homotopy category&#8221;. I begin by introducing, essentially from scratch, various functors in algebraic geometry and trying to justify that there are a number of examples of &#8220;A^1-invariant&#8221; functors. These notes are really supposed to be the first step towards an introductory text about A^1-homotopy theory that proves the affine representability of vector bundles &#8220;from scratch&#8221; building on my papers with Hoyois and Wendt.<\/p>\n<ul>\n<li>DISCLAIMER: These notes are still very much in preparation, and there are certainly mistakes and omissions. I would very much appreciate comments if you do happen to read them (and apologies to those who have sent me comments, I have not yet had time to incorporate them)!<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<h2><strong>Talk Transcripts<\/strong><\/h2>\n<ul>\n<li><strong><i>Algebraic vs. topological vector<\/i><em>\u00a0bundles<\/em><\/strong><br \/>\nNotre Dame Topology Seminar 10\/27\/2020 [<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/NotreDameTopSem.pdf\">PDF<\/a>]<\/li>\n<li><strong><i>Counting Vector bundles<\/i><\/strong><br \/>\nBrandeis-Harvard-MIT-Northeastern Colloquium 3\/9\/2017\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/BHMNColloquium.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>Vector bundles and A^1-homotopy theory<\/i><\/strong><br \/>\nAMS Summer Research Institute on Algebraic Geometry 7\/21\/2015\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/UtahTalk.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>Projective modules and A^1-homotopy theory<\/i><\/strong><br \/>\nMath. Cong. Amer, Session: Geometric aspects of homotopy theory 8\/6\/2013\u00a0<a href=\"http:\/\/www-bcf.usc.edu\/~asok\/papers\/GuanajuatoTalk.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>Connectedness in the homotopy theory of algebraic varieties<\/i><\/strong><br \/>\nUC Riverside Colloqium, 3\/31\/2011\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/Riverside.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>Rational points up to stable A^1-homotopy<\/i><\/strong><br \/>\nSatellite conference on Algebraic geometry to ICM 2010, 8\/15\/2010\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/Asok_-_HyderabadTalk.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>Rational connectivity and A^1-connectivity<\/i><\/strong><br \/>\n11th NRW Topology Seminar, 5\/8\/09\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/BielefeldTalk.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>In what sense are algebraic varieties like manifolds?<\/i><\/strong><br \/>\nUSC Colloqium, 1\/20\/09\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/USCTransparencies.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>A &#8220;homotopic&#8221; view of affine lines on varieties<\/i><\/strong><br \/>\nUCLA Colloqium, 1\/31\/08\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/UCLATransparencies.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>Unipotent groups and some A^1-contractible smooth schemes<\/i><\/strong><br \/>\nU Chicago Alg. Geom Sem, 3\/7\/07\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/UChicagotransparencies.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<\/ul>\n<h2><strong>Informal writings<\/strong><\/h2>\n<ul>\n<li><strong><i>The Jouanolou-Thomason homotopy lemma<\/i><\/strong>\u00a0(v. 9 Feb 09);\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/Jouanolou.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<li><strong><i>The unstable A^1-n-connectivity theorem<\/i><\/strong>\u00a0(v. 1 Apr. 09);\u00a0<a href=\"\/aravind-asok\/wp-content\/uploads\/sites\/213\/2023\/06\/Unstablehigherconnectivity.pdf\" target=\"_blank\" rel=\"noopener\">[PDF]<\/a><\/li>\n<\/ul>\n\n\n\n<\/div>\n\n\n  <\/div><\/div>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":370,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"footnotes":""},"class_list":["post-154","page","type-page","status-publish","hentry"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.1.1 - 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