Joint Moments of Characteristic Polynomials of Random Unitary Matrices

 I will review what is known and not known about the joint moments of the characteristic polynomials of random unitary matrices and their derivatives. I will then explain some recent results which relate the joint moments to an interesting class of measures, known as Hua-Pickrell measures. This leads to the proof of a conjecture, due to Chris Hughes in 2000, concerning the asymptotics of the joint moments, as well as establishing a connection between the measures in question and one of the Painlevé equations.