Abstract: Doubly stochastic operators are analog of doubly stochastic matrices. We will define doubly stochastic operators by using a pre-order relation that can be defined by rearrangement of functions. We discuss different characterizations of doubly stochastic operators. If time permits we will explain a new representation for doubly stochastic operators.

Abstract: G-invariant differential operators D on a homogeneous space G/H will be discussed. A principle for constructing explicit solutions to Df=0 will be explained. This will be illustrated with several examples, such as Dirac operators.