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  • 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 threefolds 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.
    • 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)!

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