Algebra/Topology Seminar

Speaker: Robert Burklund

Title: Motivic stable stems over fields

Abstract: A simple measure of our understanding of the motivic stable homotopy category of a field k is our understanding of the bigraded stable stems in that category. In the case k=C we now understand that the motivic stable stems are intimately tied with the Adams--Novikov spectral sequence and this connection has greatly enhanced our computational understanding of both classical and motivic stems. In this talk I will explain how, for fields containing a square root of -1, there are simple formulas for the p-complete motivic stable stems in terms of Milnor K-theory and stems over C. The proof of this result involves the construction of a category of cellular motives over F_1 and requires proving many new cases of the Galois reconstruction conjecture. This conjecture asserts that the subcategory of Artin--Tate objects in SH(k) depends only on the absolute Galois group of k and is now proved for fields of small cohomological dimension. This talk represents joint work with Tom Bachmann and Zhouli Xu.