The talk will describe progress over the last year on noncommutative sets and functions. One result completely characterizes the composition $r$ of a convex with an analytic noncommutative rational function, $r(z)= f(q(z))$. These are free analogues of plurisubharmonic functions.
Another result concerns free semialgebraic sets, a.k.a. free spectrahedra, oddities of optimization over them and their free extreme points.
The work is joint with Meric Augat, Erik Evert, Igor Klep, Scott Mccullough, Jurij Volcic.