Xing Gu: The ordinary and motivic cohomology of $BPGL_n(\mathbb{C})$

Abstract: For an algebraic group $G$ over $\mathbb{C}$, we have the classifying space $BG$ in the sense of Totaro and Voevodsky, which is an object in the unstable motivic homotopy category that plays a similar role in algebraic geometry as the classifying space of a Lie group in topology. The motivic cohomology (in particular, the Chow ring) of $BG$ is closely related, via the cycle map, to the singular cohomology of the topological realization of $BG$, which is the classifying space of $G(\mathbb{C})$, the underlying Lie group of the complex algebraic group $G$. In this talk we present a work which exploits the above connection between topological and motivic theory and yields new results on both the ordinary and the motivic cohomology of $BPGL_n(\mathbb{C})$, the complex projective linear group.