To analyse a statistical problem, I always try to use its geometry. This is perhaps most apparent in the mixed primal-dual bases algorithm that I developed for concave programming. But it is also apparent in the work on the decomposition of the li kelihood ratio statistic for testing the homogeneity of the covariances in several given samples from multivariate distributions. To analyse the structure of a problem, one can also put the problem in an appropriate algebraic setting. The Wishart distribution has many properties that are most efficiently analysed in the framework of Euclidean Jordan algebras and symmetric co nes.
Symmetric cones are nothing but the generalizations of the cone of real positive definite matrices.
Wishart distributions are essential in the study of the traditional covariance selection model and the recently developped graphical models. Using standard methods of the theory of symmetric cones has allowed us, for example, to find an explicit closed form formula for the covariance of certain graphical models. Most of my current research is in the area of the Wishart distribution and graphical models.