Free functions preserving certain partial orders of operators

Recently free analysis has been a very active topic of study in operator and function theory. In particular free functions that preserve partial orders of operators have been studied by a number of authors, in connection to Loewner's theorem and interpolation problems on the polydisk. Also operator concave free functions naturally get into the picture as we study the positive definite order preserving free functions. We will go through recent results in the field, and we will cover some recent works on analytic lifts and extension of operator monotone and concave functions to the matrix convex hull of their domains. This is related to some conjectures in the field, for instance McCarthy's conjecture that we can address now. If time permits, we will cover another recent joint work with M. Gaál solving Blecher's problem on characterizing real positive definite order preserving functions.