Oliver Röndigs

John Milnor's attempt at the higher algebraic K-groups of a field 
uses the field units as generators and the Steinberg relation as 
the only relation. The resulting Milnor K-groups map to the higher 
algebraic K-groups Daniel Quillen defined slightly later as homotopy 
groups of a certain topological space, and the map is an isomorphism 
in degrees 0, 1, and 2. A decade later Andrei Suslin constructed a 
Hurewicz-type homomorphism from Quillen's algebraic K-groups to 
the Milnor K-groups of a field and proved that the resulting endomorphism 
on the n-th Milnor K-group is multiplication by (n-1)! if n>0. 
He conjectured that the image of the Hurewicz-type homomorphism 
is the same as the image of this endomorphism (hence as small as possible) 
and proved the degree 3 case. Recently Aravind Asok, Jean Fasel, and Ben 
Williams ingeniously used $\mathbb{A}^1$-homotopy groups of algebraic spheres to settle 
the degree 5 case. Also using $\mathbb{A}^1$-homotopy groups, but now for the projective 
plane, I will explain how to treat the fourth degree.