Cellular A1-homology is a new homology theory for smooth algebraic varieties over a perfect field, which is often entirely computable and is expected to give the correct motivic analogue of Poincaré duality for smooth manifolds in classical topology. I will introduce cellular A1-homology, describe the precise conjectures about cellular A1-homology of smooth projective varieties and discuss how they can be verified for smooth projective rational surfaces. The talk is based on joint work with Fabien Morel.[-]

The loop Grassmannians of reductive groups will be reconsidered as a construction in the setting of “local spaces” over a curve. The notion of a local space is a version of the fundamental structure of a factorization space introduced and developed by Beilinson and Drinfeld. The weakening of the requirements formalizes some well-known examples of “almost factorization spaces.” The change of emphases leads to new constructions. The main example will be generalizations of loop Grassmannians corresponding to quadratic forms Q on based lattices. The quadratic form corresponding to the loop Grassmannian of a simply connected group G is the basic level of G.[-]