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  1. ; (, Proceedings of the International Congress of Mathematicians)
    Beliaev, D; Smirnov, S (Ed.)
    We survey recent developments related to the problem of classifying vector bundles on algebraic varieties. We focus on the striking analogies between topology and algebraic geometry, and the way in which the Morel–Voevodsky motivic homotopy category can be used to exploit those analogies. 
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  2. ; ; (, International Mathematics Research Notices)
    Abstract We analyze the question of which motivic homotopy types admit smooth schemes as representatives. We show that given a pointed smooth affine scheme $$X$$ and an embedding into affine space, the affine deformation space of the embedding gives a model for the $${\mathbb P}^{1}$$ suspension of $$X$$; we also analyze a host of variations on this observation. Our approach yields many examples of $${\mathbb A}^{1}$$-$$(n-1)$-connected smooth affine $2n$-folds and strictly quasi-affine $${\mathbb A}^{1}$$-contractible smooth schemes. 
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