I will review the recent mathematical approach to Conformal Field Theory proposed by my colleagues Nam-Gyu Kang and Nikolai Makarov. Their construction defines a certain class of algebraic operations on correlation functions of the Gaussian Free Field, and these operations can be used to give meaning to "vertex observables" and other well known objects in CFT. Using conformal transformation rules for the GFF these objects can be defined on any simply connected domain, and using Lie derivatives they can be analyzed when the domain evolves according to an infinitesimal flow. Using the flow of Loewner's differential equation produces a connection with the random curves of the Schramm-Loewner evolution, which I will describe along with some recent work in the case of multiple SLE curves.

We continue the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We have established some relations with forcing axioms and with the existence of certain regular forcing embeddings and projections, and also point out connections with Precipitousness. We give an rough overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory. In this talk we explain one of the main technique used in the argument, namely the frequent extension argument.

We continue the exposition on self-genericity axioms for ideals on P(Z) (Club Catch, Projective Catch and Stationary Catch). We establish some relations with forcing axioms and with the existence of certain regular forcing embeddings, and also point out connections with Precipitousness. In particular we observe that if Projective Catch holds for an ideal, then that ideal is precipitous, and the converse holds for ideals that concentrate on countable sets. Finally we give an overview of the method used for proving the existence of models with Woodin cardinals coming from these axioms, using the Core Model Theory.