The interplay between structural  Ramsey theory and topological dynamics of automorphism groups has been extensively studied since their connection was established in a paper by Kechris-Pestov-Todorcevic, while earlier works of Pestov, and Glasned and Weiss exhibited the phenomena in special cases. This line of research was extended to metric structures and approximate Ramsey property by Melleray and Tsankov. We establish the approximate Ramsey property for the class of finite-dimensional normed vector spaces and deduce that the group of linear isometries of the universal approximately homogeneous Banach space, the Gurarij space, is extremely amenable, that is, every continuous action on a compact Hausdorff space has a fixed point. Dualizing our ideas, we show that the class of finite-dimensional simplexes with a distinguished extreme point and  affine surjections satisfies the approximate Ramsey property. As a consequence, we find that the universal minimal flow of the group of affine homeomorphisms of the Poulsen simplex is its natural action on the Poulsen simplex. This is a joint work (in progress) with Aleksandra Kwiatkowska (UCLA), Jordi Lopez Abad (ICMAT Madrid and USP) and Brice Mbombo (USP).

We finish the proof of Shelah's 'Trichotomy' theorem.

We introduce some of the basic objects in PCF theory like good (flat) points and exact upper bounds for sequences of functions in reduced products of regular cardinals. We will then give a proof of Shelah's 'Trichotomy' theorem.

Baumgartner's Axiom postulates that any two $\aleph_1$-dense subsets of the real line are order-isomorphic. A set is $\aleph_1$-dense iff every nonempty open interval intersects the set in $\aleph_1$-many points. We present Todorcevic's argument which shows that Baumgartner's Axiom is a consequence of the Proper Forcing Axiom.

Woodin's $P_{max}$ forcing when applied to a model of Determinacy produces a model which is maximal for sets of countable ordinals. We will briefly introduce $P_{max}$ and its applications and variations, and outline a proof of the maximality of $P_{max}$ extensions.