Principal components analysis is a staple of multivariate statistical 
analysis. Viewed as the study of the eigenvalues of a sample covariance matrix, it is 
an important example for random matrix theory. The talk will explore the interplay 
between these two subjects by focusing on covariance matrices which are drawn 
from low rank perturbations of a scaled identity matrix. Such models arise in settings 
as diverse as finance, genetics and signal processing -- brief examples will be given. 
We give an overview of some results of several people on estimating and testing in 
settings with both weak and strong signals. 
