• Vector bundles on algebraic varieties (with Jean Fasel); ICM 2022 [YouTube]
  • Vector bundles and A^1-homotopy theory; 2015 AMS Summer Algebraic Geometry Program [YouTube]

Extended abstracts

  • Rational points vs. 0-cycles of degree 1 in stable A^1-homotopy; with C. Haesemeyer and F. Morel
    (v. 16 Jun 2010) Oberwolfach Reports Volume 7, Issue 2, 2010 pp. 1423-1425 [PDF]
  • Toward a metastable range in A^1-homotopy theory of punctured affine spaces; with J. Fasel
    (v. 22 Jul 2013) Oberwolfach Reports Volume 10, Issue 2, 2013 pp. 1892-1895 [PDF] 
  • On the Freudenthal suspension theorem in unstable motivic homotopy theory; with T. Bachmann and M.J. Hopkins (v. 1 Jun 2022) Oberwolfach Reports 2022 [PDF]

Lecture Notes

  • A^1-contractibility and topological contractibility – [PDF]
    Notes for lectures at the Fields Institute workshop entitled “Group actions, generalized cohomology theories and affine algebraic geometry” at U. Ottawa

    • An affirmative answer to Question 2.4.5 was given by A. Dubouloz and J. Fasel in their paper Families of A^1-contractible affine threefolds. The work of Dubouloz and Fasel was motivated in part by the work of M. Hoyois, A. Krishna and P.-A. Ostvaer entitled A^1-contractibility of Koras-Russell threefoldswhich 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.
    • 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.
    • (Update: 12/21/2021) Further advances can be found in A^1-homotopy theory and contractible varieties: a survey written jointly with P.-A. Ostvaer.
  • Algebraic geometry from an A^1-homotopic viewpoint – [PDF 2016 version] [PDF 2021 version]
    Notes 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 “naive A^1-homotopy category”. I begin by introducing, essentially from scratch, various functors in algebraic geometry and trying to justify that there are a number of examples of “A^1-invariant” 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 “from scratch” building on my papers with Hoyois and Wendt.

    • 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)!

Talk Transcripts

  • Algebraic vs. topological vector bundles
    Notre Dame Topology Seminar 10/27/2020 [PDF]
  • Counting Vector bundles
    Brandeis-Harvard-MIT-Northeastern Colloquium 3/9/2017 [PDF]
  • Vector bundles and A^1-homotopy theory
    AMS Summer Research Institute on Algebraic Geometry 7/21/2015 [PDF]
  • Projective modules and A^1-homotopy theory
    Math. Cong. Amer, Session: Geometric aspects of homotopy theory 8/6/2013 [PDF]
  • Connectedness in the homotopy theory of algebraic varieties
    UC Riverside Colloqium, 3/31/2011 [PDF]
  • Rational points up to stable A^1-homotopy
    Satellite conference on Algebraic geometry to ICM 2010, 8/15/2010 [PDF]
  • Rational connectivity and A^1-connectivity
    11th NRW Topology Seminar, 5/8/09 [PDF]
  • In what sense are algebraic varieties like manifolds?
    USC Colloqium, 1/20/09 [PDF]
  • A “homotopic” view of affine lines on varieties
    UCLA Colloqium, 1/31/08 [PDF]
  • Unipotent groups and some A^1-contractible smooth schemes
    U Chicago Alg. Geom Sem, 3/7/07 [PDF]

Informal writings

  • The Jouanolou-Thomason homotopy lemma (v. 9 Feb 09); [PDF]
  • The unstable A^1-n-connectivity theorem (v. 1 Apr. 09); [PDF]