#### Abstract

The classifying space of the nth projective unitary group, BPUn, is a fundamental mathematical object. However, unlike its close cousins BUn, BSUn, etc., surprisingly little has been known to its cohomology. I will briefly introduce the roles of BPUn in the study of the topological period-index problem and anomalies in theoretical physics, and present recent progresses on the computation of its cohomology.

In the context of algebraic geometry, BPGLn, the classifying stack of the projective linear group PGLn, is an analog of BPUn, while the theory of Chow rings plays a similar role as singular cohomology. I will point out where the above computation pass to the Chow ring of BPGLn.