Even spaces and motivic resolutions

Abstract: In 1973 Steve Wilson proved the remarkable theorem that the even spaces in the loop spectrum for complex cobordism have cell decompositions with only even dimensional cells. The (conjectural) analogue of this in motivic homotopy theory leads to a surprising resolution of cellular varieties into motivic complexes. In this talk I will survey this situation and describe joint with with Mike Hill and with Jean Fasel and Aravind Asok progress and on potential applications.