We complete the discussion of Baumgartner's axiom and related topics.

It appears to be an open question whether for every regular uncountable regular $\lambda$, every automorphism of $P(\lambda)/fin$ is trivial on a co-countable set. We will show that a small fragment of Martin's Axiom implies that if $\lambda$ is at most the continuum then every automorphism of $P(\lambda)/fin$ which is trivial on sets of cardinality less than $\lambda$ is trivial.
 

We discuss the history of Baumgartner's result that all \aleph_1-dense sets of reals can be order-isomorphic, as well as related results of Shelah and Abraham. We'll outline a proof, due to Todorcevic, that is simpler than Baumgartner's original argument. Finally, we present some recent results of Justin Moore concerning the problem of making all \aleph_2-dense sets of reals isomorphic.
 

We complete 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 the mechanism of absorbing extenders in the core model, and lifting iterability from countable models to models of large cardinality.

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 the mechanism of absorbing extenders in the core model.