# Jonas Kylling: Algebraic cobordism of number fields

I will report on work in progress on calculations of the motivic homotopy groups of MGL (the algebraic cobordism spectrum) over number fields. It is known that pi_{2n,n}(MGL) is the Lazard ring, and pi_{-n,-n}(MGL) is Milnor K-theory of the base field. We will calculate all of pi_{*,*}(MGL) with the slice spectral sequence (motivic Atiyah-Hirzebruch spectral sequence) over a number field. I will give a brief review of the the tools and sketch the main parts of the calculation: The input from motivic cohomology, the use of C_2-equivariant Betti realization and comparison with Hill-Hopkins-Ravenel to determine the differentials, and settle most of the hidden extensions.