Random Matrices with Prescribed Eigenvalues

Classical random matrix theory begins with a random matrix model and analyzes the distribution of the resulting eigenvalues.  In this work, we treat the reverse question: if the eigenvalues are specified but the matrix is "otherwise random", what do the entries typically look like?  I will describe a natural model of random matrices with prescribed eigenvalues and discuss a central limit theorem for projections, which in particular shows that relatively large subcollections of entries are jointly Gaussian, no matter what the eigenvalue distribution looks like.  I will discuss various applications and interpretations of this result, in particular to a probabilistic version of the Schur--Horn theorem and to models of quantum systems in random states.  This work is joint with Mark Meckes.